Birmingham gambit against the Polish opening 1.b4 c5!?

white is under no obligation to play this in normal orangutan style and he prob shoudnt

whether you go bb4+ or be7 , white is slightly better thanks to the d5 isolani. white will go bd3, ne2 0-0, rb1,rc1 and have a good game

Only 1% of players play c3, and the position is objectively equal.

Anyway, if you don't like the resulting position, Black can give up the pawn with a dynamic position

+= is not "equal" , besides, why give black what he wants by taking on e5?

this is a very pleasant french position, the d5 pawn has no c6 to protect it, and the "bad french bishop" can go to ba3. This is also slightly better for white. 
you keep giving these dumb statistics, When you prepare an opening line, you always do so from the perspective that your opponent will play the critical lines ,it doesnt matter if its seen 15% or 1% of the time, when people prep agaisnt those lines, the very first thing they will do is open the engine and check what it thinks its most critical, the 2nd thing they will do is look for secondary alternatives that even if not absolute best, spoil your plans.

There is literally 5 master games in the database with 1.b4 c5, after bxc5 e5, there is no consensus among those games how to proceed. but, all reasonable continuations give white above average advantage, compared to the 1.b4 main lines. There is no particularly good reason to try this. you literally let white get a favorable queen pawn game (or an even more pleasant reversed french) by slightly misplacing the bishop and isolating the d pawn for not much.

You have absolutely no clue how openings are prepared.

Your only goal is to contradict me.

Besides, I don't know what you're doing looking at the master database.

Check the Lichess database with 6 billion games and then tell me.

Almost nobody plays c3, and it's not that big of a deal.

This isn't a line that leaves you lost, so there's no reason to stop playing it.

But you're implying exactly that with your stubbornness.

Anyway, your line is still playable for Black.

The key is not to capture the pawn or push it to e4 just yet

only when it's actually worth it

White is almost forced to play Nxe5, leading to a slight advantage due to the extra pawn, but objectively, the position is very playable for black

almost anything is "playable" vs 1.b4. When accessing an opening line, both the objective and relative value of a move is to be considered. a line can objectively be 0.2 , if the critical line is 0.8, it is a inferior move. Here black has given up half a dozen ways to get instant equality and handed over the advantage back for...not much.

wait, you mean to tell you me you use the database thats mostly rapid and blitz games from unknown players to base your opening decisions? the main use of the lichess game database is to check what is most often played as a first instinct by decent players esp when the master database is running thin. If you want to base your bullet repertoire on lines which have a high chance to fall to an opening trap, that is fine, otherwise, this is hope chess. This reflects the folly of thinking only what lets you win matters which is paradoxically both true and an impossible mentality to keep if you want to seriously improve. Chess is too objective a game to get far like that.

thwarting you has never been my primary ambition i can promise you that. the self sabotage does all the heavy lifting for me.

I am already retired from classical chess, and I have also mentioned in some threads that in slow games of 2 hours, it is preferable to have a more solid approach because playing gambits is heavily punished there. Playing in such a risky manner is penalized a lot, at least in the long run.

I started drawing and winning against masters in classical chess the day I abandoned gambits and began playing the Modern Defense and the Caro-Kann, for example.

I simply changed my approach, but in blitz, I still say the same thing: it is more profitable to play hyper-aggressively with this c5 rather than choosing lines where White is more accustomed to playing.

Taking my life in my hands here, heading into the Big Boys' corner of the playground, but ahh what the heck?

I occasionally play 1. b4 when feeling puckish, but can't call myself an expert. My immediate instinct is to play 2. b5 against 1..., c5; based on nothing more than the General Principle that White wants to try and hit c6 preventing or inconveniencing the smooth development of the Nb8 and maybe augmenting a fianchettoed Bg2 if thing go that way. And 1..., c5 prevents Black from challenging the Pb5 with ..., c6.

My next step? Pop to a database and an engine to see whether/to what extent my instinct is awful. I'll confess what I find here...unless some far better player such as the two who have already posted here beat me to it with a critique.

Fascinating. The database I use suggests White usually misplays the resulting position and tries to keep a Pawn on b5 by creating a Pawn Chain. With grim results. Meanwhile, Stockfish agrees with my instinct that the general White plan should involve an early e3, then heading for a St. George type formation...initially assessing it as better for Black then changing to fractionally better for White (as close to equal as makes no difference) after a few more moves played on the board. So I'd agree that 1..., c5 is a nicely practical idea. Especially at quick time limits.

It seems that the best option for Black is to play a6 against that b5 to at least trade off a pawn, though there are many playable options.

At levels 2200 and above, White captures the pawn on c5 about 80% of the time.

9% of the time, they play your b5 move, and another 9% go for Bb2, which seems dubious.

b5 is fine, and I think I would play a6, as I said—at least trading a pawn to make Black's game easier.