This is a cool side line, and one that doesn't seem to have much analysis out there. It strikes me as having similar flavours to the Englund Gambit, although I'm sure it's more sound and has greater chances for complications.
Look forward to seeing some games!
My first 3-0 game went like this lol:
Seems like this will offer a few free wins...
By the way, I was analyzing the 6...Qxd4 line and it seems black has an interesting resource with 8...Bc5!:
Accurate play is necessary here, but black can claim a lot of compensation for dropping the rook on a8.
LMAO @ your first game!
You're right regarding Black's resources when he snags the other pawn (it gets very sharp). For what it's worth I've never once encountered it.
I neglected to mention it before, but I think the main reason I like this line is that Black's play is far more predictable (less deviations) than in the traditional BDG lines. So, less to memorize and you get a lot of practice in the same positions. In short time controls this has paid off for me... though I admit that I still end up losing games despite grabbing a piece... like the game below, ha
Haha. I don't think the result of that game had much to do with the opening 
.
And I mostly agree that the von Popiel seems to involve memorization of less variations than the BDG. The most trying defense for black seems to only be 4...Bf5 in the von Popiel, whereas black has many options after 5.Nxf3 in the BDG.
I think it's worth studying the Burn variation of the French defense before playing this though, as there will be a strong chance of the position transposing after 4...e6 5.Nxe4.
True. Most my games either continue with BDG or transpose to French. I don't mind the Burn variation, it's more open than the typical locked structure of the French and I do better in open positions. Aside from that variation, I usually play the Fingerslip variation (fun!) or the Alekhine-Chatard (also fun).
Haven't logged on in a while, but this is the variation I always used to play. I believe there a bunch of games from David Gedult with this opening (rarely best play from either side though).
And my favorite part of the BDG or veresov is transposing into a French while avoiding all the other 1.e4 lines. Cyrus Lakdawala includes some nice anti-french lines in is Ferocious Opening Repertoire.
About five years later and I'm still exclusively playing this opening as White.
I'm familiar with most all the side lines and deviations. I still lose spectacularly from time to time, but I'm able to blitz out the first 10 moves without thinking while my opponent burns their time. I've won games on the clock that I was completely losing b/c of this. Yes, a dirty flag, but that's part of the game.
I've been playing the BDG more seriously now for a number of months (on another chess site) and I've focused on the von Popiel Gambit. It is as follows:
Computer evaluation shows this as inferior to the 4. f3 variation. But I like it better because the moves your opponent will play are super predictable. One downside is that your opponent may choose to play 5. ...d6 to transpose into the French. I don't mind this because I have several aggressive French variations that I like to play. You can play 6. f3 at this point if you're committed to the BDG, but I think there are more aggressive lines in the French.
It is literally SO EASY for your opponent to drop a piece in this version. Here is a trap I have pulled off so many times I can't even count... mostly resulting in instant resignation by Black (because they are so pissed-off that they fell for it haha).
Your opponent might get extra greedy. Bad news if they do. There are some ways for Black to limit the damage but very hard to find in a game with little time and unknown territory. This is a very common scenario that I've had plenty of times:
I encourage you to give this variation a shot. There are a ton of traps, and even if Black plays some of the better moves, you get a nice attacking, sharp game. For example:
I'm not saying the above is better for White. It's not. It's pretty fun to play though and asks Black all the questions.
Anyway, I'll look to post a few example games sometime. For now, I encourage you to spend a week or so playing this variation. You may never go back to the typical BDG lines