1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+ Ke7
This is how my most recent opponent played. Declining the offer of the Bishop is rare, and doing so by moving the King to e7, instead of f8, is even more rare.
5.Bxg8
This was a suggestion from my last post, and it should be considered, if only because it "solves" the issue of where, otherwise, to retreat the Bishop.
5...Qxg8
The correct recapture, but only seen in 1 of 8 games in The Database. The problem with 5...Rxg8, seen as early as Drewbear - AAlekhine, Jerome Gambit thematic tournament, ChessWorld.net, 2007 (1-0, 33) is that White replies 6.d4, threatening to skewer Black's King and Queen with 7.Bg5+, thus winning a piece.
Alas, only two players responded correctly to 5...Rxg8 - Drewbear, in 2007, and Bill Wall, in 2015 - the latter, the game ended after 10 moves.
As for the text, I have only one game to guide me, a lightning game from 2008. We can follow that for a bit.
6.d4
This is kind of funny. Stockfish 10 prefers 6.Nc3 d6 7.Nd5+ Kd8 8.d4 exd4 9.Bg5+ Kd7 10.0-0, giving up a pawn to do - what? He develops his Bishop, forcing Black's King to block Black's light-squared Bishop, which, in turn, blocks Black's Rook... Certainly a Jerome Gambit strategy if ever there was one.
6...exd4
7.Bg5+
Striking at the King, although, with the Black Queen at g8, the move is not as strong. Probably 7.c3 was better, when Black could try 7...dxc3 or 7...d5. White would have the safer King, but he would be playing quite a different game than the usual Jerome Gambit.
7...Kd8 8.0-0
Reasonable, although the computer recommends 8.c3.
8...Be7
Simply allows White to recover his pawn.
9.Bf4
Fine, but 9.Bxe7 Kxe7 10.Nxd4 was hard to argue with.
9...Qf7
Now a tactical skirmish advanced White's game.
10.Bxc7 d6 11.Nxd4 Kd7 12.Nxc6 Kxc7 13.Nxe7 Qxe7 14.Re1 Rf8
Here, the clock seemed to step in.
15.e5 Qh4 16.Qe2
Instead, 16.Qxd6 was checkmate, but, remember, this was a bullet game. White won at move 32.
[This is blog post #2,750. Thanks for reading!]