That is a strange question. If white can take back with the rook, he simply has a winning position: Healthy extra pawn and bishop pair. Besides, the rook is perfectly well placed to support the pawn break c4-c5. The pawn break that will not be possible anymore after the Bxc3 bxc3 exchange, as Phren already pointed out.
Considering the Budapest

@ FM Rumo
Don't get overexcited! You have a lot to do, to win this position. A c pawn brake does not secure a victory because white has the bishop pair! Go win this position OTB and then come and tell us how easy it is.

I played the opening long enough to understand that 6...Bxc3+ gives nothing. I know that the Budapest is for romantic souls but i didn't knew that serves and illusionist like you. The double c pawns is not a weakness for white and black has absolutelly no plan to create considerable counterplay. Perhaps instead of wards you want to show something from your rich experience that everyone on this planet missed!?
I read Moskalenko's book and i don't see nowhere a concrete plan for black after 6...Bxc3+ or 7...Bxc3+. In order to sell the book more he didn't mention not even one from Gligoric games in the Budapest Rubinstein variation. Anti-selling example.
Doubled isolated c-pawns and an isolated a-pawn aren't weakness for White? Um, ok. I suppose the correct way is to take on c3 without wrecking White's pawns as you showed in your diagram, that makes perfect sense.
BxNc3+ is a very basic and key move for Black in that line. If you don't grasp why that move is good you have no business even studying openings, you need to go read "Basic Strategy for Beginners" before you run your mouth off about anything, much less an opening you don't even play. (Oh but you used to play it, and gave up on it before you learned it past move 5, I know this makes you an expert about it.)
And apparently you didn't read Moskalenko's book as you claim (or else you have zero reading comprehension) because he shows lots of ideas, in particular, the Schlechter Knight manuever of Nf6 to e4 to c5 blockading the isolated doubled pawns. This is elementary strategy straight out of My System. Meanwhile White's queen is exposed in the center, Black has a lead in development, semi-open e and f files, and the knight on c6 can hop to e5 creating what Taylor referred to in his book as "Schlechter's double-fisted knight blockade", all this ON TOP of the shattered White queenside and weaknesses to work with adds up to considerable counterplay. This is the price White pays for keeping the extra pawn and why lots of players prefer Nbd2.
While a weakness may be there is it actually exploitable? Black castles kingside in the Sveshnikov despite the open king because theory knows white can't make use of it unless black is asleep. The strong knight activity is why I'm considering the Budapest, it's nasty dealing with solid knights planted on your side of the board or near it. Then again I get that with the Leningrad Dutch (...Na6-...Nc5 and eventually ...Ne4 with support from the king's knight and f-pawn is theory) but on the other hand there are nasty sidelines (1.d4,f5 2.Nf3,g6? 3.h4!,Nf6 4.h5!,Nxh5 5.Rxh5,gxh5 6.e4 white has serious compensation for the pawn, so I prefer instead 2...d6!? but 3.h4!? discourages a Leningrad setup)
The Grunfeld looks great, but has lots of counterintuitive moves you need to know at critical junctures to avoid giving white a clear advantage.

There is no extra pawn if black castles first! This is my point.
Perhaps IM Pfren would like to show us why this position considers much inferior for black? Thank you.
Bishop_g5, not immediately capturing on c3 also gives White the option of 7.Qb3, when the pawn can't be recaptured because the bishop would hang and if the bishop takes on c3 the queen recaptures and defends e5.
Let me make this as simple as I possibly can for you. You have two options 1. Take on c3 at move six and wreck White's queenside structure, or 2. Wait around and only take on c3 after White arranges to recapture with a piece. For some reason none of the rest of us understand you're advocating option 2.
While a weakness may be there is it actually exploitable?
The queenside weaknesses will be exploitable sooner or later. This isn't one of those things like "it's not a real weakness because it can't be attacked", the entire queenside is smashed and the weaknesses are pretty glaring. White will have to try and keep pieces on, and when Black wins one of the pawns back White is just going to be worse because although the material will then be equal White will still have the worse pawn structure. Also in positions with doubled isolated pawns a knight tends to be better than a bishop so the lack of bishop pair isn't such a big deal here.
Whatever opening you select I just hope you stick with it. If you go with the Budapest there may be more of a temptation to give up on it because people will be telling you it sucks all the time.

And what makes 7...Ba5 inferior?! Tell me why Rxc3 gives an initiative to white and I will agree with you. Yes he has the c pawn brake: it's enough? Yes he has the bishop pair: it's enough?
Dolphin listen to me...I am not trying to be stubborn here. Just someone to show me why? Seirawan didn't, he showed a potential trap for white to capture the Queen and spoke for whites Queen side expansion where I don't see how this leads to an advantage all the way. There are no books that writes about this position or any game where you can see and say, oh yes! In this variation black has no response to defend the position. Nothing! I hate the stereotypes in chess when nothing is proven. Enough with this!
From my perspective I wouldn't like to play a move like 7...Ba5 for the same reason I don't like Taylor's recommendations of keeping the bishop pair in the Nbd2 Rubenstein. Black is going to fall behind in development making all these repetitive piece moves. We're already moving our kingside knight three times and then if you start moving the king's bishop several times it just becomes too much. I suppose if Black really wants to they can keep the pawn in the Nc3 Rubenstein just as they can keep the bishop pair in the Nbd2 Rubenstein, but at what cost? And why? If you can have a position as Black where White is slightly better and material is equal or White is only slightly better but up an entire pawn why would you want the first one? After capturing on c3 and doubling the pawns it's almost like you've gotten your pawn back already.

@ Dolphin and Ponz111
Look guys, I don't speak radomly here. After black decline the capture on c3 and play 6...Qe7 the preferences from experts and professional to play 7.Rc1 it's a fourth choice, just go and look any data you want! That shows that the position with Rxc3 and 10...0-0 it's not ideally either for white and requires some kind of preparation. The bishop pair and the slight initiative on the Q side guarantees nothing.
The first preference from white after 6...Qe7 remain the 7.Qd5 because the psychology of white player after the Bf4 move is to do not abandon the e pawn. When white plays Bf4 this is what he wants. The second option of Qb3 it's an attempt to force the bishop exchange with the knight and at the same time to keep an extra eye on e pawn but after 7...Ba5 white has nothing and black takes back his pawn while he's development is more completed than white is!! The third option of 7.e3 it's much more interesting because white accept rather slowly that chasing the e pawn has no meaning and open the L Square bishop diagonal to continue his development.
My point is that we do a favor White by capturing immediately the c3 knight when what he wants is to keep focus on e pawn! What black wants is to find tactical spots to gain an advantage with all pieces on the board not with Nimzo-plans.

@ achja Before we reach on that white goes Qd5 with the idea to force ...f6 and exchange his gambit, so don't tell me about deeper ideas. Meanwhile the Schlechter knight traveler does not allow c4-c5 conservatives. So...in the end white need that e pawn worthwhile because the deeper plan might not occur.
This is a fact: The Bf4 variation promises Qd5 with or with out deeper ideas.

@ Dolphin and Ponz111
Look guys, I don't speak radomly here. After black decline the capture on c3 and play 6...Qe7 the preferences from experts and professional to play 7.Rc1 it's a fourth choice, just go and look any data you want! I never said it was not fourth choice. Was just commenting on one position and showing how it was much better for White.
That shows that the position with Rxc3 and 10...0-0 it's not ideally either for white and requires some kind of preparation. The fact that it is 4th place does not necessarily mean anything, it could be the 4th place is really the best follow up.
The bishop pair and the slight initiative on the Q side guarantees nothing. Wrong! You are completely wrong here! What you are doing is to gloss over White has a slight material advantage of the 2 bishops vs bishop and knight and the iniative of the queenside. So White has 2 advantages and though each one, in itself, might be slight, the two together add up to a very nice advantage for White and a position where Black is on the defensive and can only hope [and it is only a hope] to somehow defend for a long period of time and get a draw.
The guarantee is that White gets a very good game with a high percentage hope of winning and an easy game to play and Black is guaranteed nothing except to strive very hard for many moves to try and draw a very inferior game.
The first preference from white after 6...Qe7 remain the 7.Qd5 because the psychology of white player after the Bf4 move is to do not abandon the e pawn.Don't care what you believe is "the first preference" what we were discussion was one chess position!
Don't care what you think the psychology of the white player is after Bf4. What you need to understand is that a good player does not just think, "I will hold on to the pawn" NO! he thinks, "I may or may not hold on to the pawn--if I can get a good position without holding on to the pawn, then that is, for sure!, an option.
When white plays Bf4 this is what he wants. Nope, wrong! He wants to win or get a good position. If he gets a good position by holding on to the pawn or not is not relevant.
The second option of Qb3 it's an attempt to force the bishop exchange with the knight and at the same time to keep an extra eye on e pawn but after 7...Ba5 white has nothing and black takes back his pawn while he's development is more completed than white is!! Does not matter [and I am not at all sure you are correct] I was referring to one position--not this.
The third option of 7.e3 it's much more interesting because white accept rather slowly that chasing the e pawn has no meaning and open the L Square bishop diagonal to continue his development. I do not care what is interesting to you. I was referring to one position.
My point is that we do a favor White by capturing immediately the c3 knight when what he wants is to keep focus on e pawn! What black wants is to find tactical spots to gain an advantage with all pieces on the board not with Nimzo-plans. Don't really care about what your point of view is. This has nothing to do with the position, I showed and discussed.
I don't think the Budapest is a bad opening, but it seems to me that in the Bf4 + Nbd2 line white can secure a safe risk-free += thanks to his bishop pair, without too many active possibilities for black.
I think you mean 5...Bb4+ like in this game: [...]
Actually I meant exactly what I said.

@ Ponz 111
Sorry, but you never showed how is much better for white! In order to win someone a won position must have a plan and a method of tactics to achieve that. I didn't see anything of that in your diagram except from perspectives and predictions. Give me something...
There is no extra pawn if black castles first! This is my point.
Perhaps IM Pfren would like to show us why this position considers much inferior for black? Thank you.
I don't like it when people shift the goalposts. Earlier you said:
"Seirawan spoke because white has 7.Rc1 to recapture then with the rook but Rxc3 has some meaning only if black plays d6 before castles, because if he 0-0 first then I don't know what white should do with the rook on c3?! even with one pawn up."
Obviously with one pawn up, this is a clear winning position. And as Ponz correctly pointed out, even after giving back the pawn white has a stable advantage. It should be added (repeated) that the rook on c3 is just perfectly placed for white's Q-side play, which is mainly based on c4-c5.
@ Ponz 111
Sorry, but you never showed how is much better for white! In order to win someone a won position must have a plan and a method of tactics to achieve that. I didn't see anything of that in your diagram except from perspectives and predictions. Give me something...
It does not need to be "shown" that 2 bishops are better than bishop+knight in this structure.

@ Ponz 111
Sorry, but you never showed how is much better for white! In order to win someone a won position must have a plan and a method of tactics to achieve that. I didn't see anything of that in your diagram except from perspectives and predictions. Give me something...
I am sorry you do not "see" the big White advantage and think someone has to "show" you how to play the position.
This is your problem, not mine.

@ FM Rumo
I didn't shift my post, when I wrote the " even with a pawn up " I was remembering a variation where black loses the d pawn but it is not necessary...lol
@ IM Pfren
Ok...I agree that white in 7.Rc1 line has an advantage. Depends from the strength of white player this can be devastated for black or playable, I understand that. I can't explain why is fourth choice in preferences and I can't blame 6...Bxc3 because I didn't found so far how to win with this! The Budapest gambit remains a mystery in to what understand about it.
@ Ponz111
You are right, but if you don't ever have play a opening position , you can't see fifteen moves ahead. Experience make someone better, not facts.
If a GM ( Miezis Normuds ) plays it and win with that, and also quite few more then I am not so...alone.