Blackburne Shilling Jerome Gambit: Queen or King?

In the following rapid game, Black is soon faced with an impossible choice.

ryuugu-rena - juen1234

10 0 rapid, lichess.org, 2023

1.e4 e5 2.Nf3 Nc6 3.Bc4 Nd4 

Named the Blackburne Shilling Gambit, although I have not been able to find a Joseph Henry Blackburne game with this line.

4.Bxf7+ 

Named the Blackburne Shilling Jerome Gambit, although I have not been able to find an Alonzo Wheeler Jerome game with this line.

The Database has 7, 496 games with this position, with White scoring 57%.

ryuugu-rena has done a bit better: in 65 games, he has scored 65%.

4...Kxf7 5.Nxe5+ Ke6 

Bypassing the somewhat more popular 5...Ke8, this time.

The Database has 3,001 games with 5...Ke8, with White scoring 59%. 

Again, ryuugu-rena has done better: in his 24 games with 5...Ke8, he has scored 67%.

As for the text move, 5...Ke6, The Database has 2,133 games with White scoring 54%.

ryuugu-rena has played 5...Ke6 18 times, scoring 64%.

6.f4 

This move loosens White's Kingside.

Instead, Stockfish 15.1 recommends (30 ply) 6.c3 Nc6 7.d4 Nxe5 8.dxe5 d6 9.Qb3+ with an edge for Black. 

6...Bd6 

Hoping to eject the enemy Knight, but it is a costly mistake.

Taking advantage of White's last move would have been 6...Qh4+ 7.g3 Qh3 8.Nf3 Nxf3+ 9.Qxf3 Kf7 (Stockfish 15.1, 30 ply) when the first player would not have compensation in terms of attack for his piece-for-2-pawns sacrifice.

7.Qg4+ Ke7 

Alas, 7...Kf6 would lose his Queen to 8.Qg5+ Ke6 9.Qxd8. 

However, this move loses his King.

8.Qxg7+ Ke6 9.Qf7 checkmate