4.Nd4 Qf6 5.c3 Nc6 looks alright to me, as 6.Nxc6 (the computer suggests 6.d3, which appears to be untested) dxc6 7.Qh5+ g6 8.fxg6 hxg6 (Downey-Bartsch, email 2002) looks OK for Black.
4.Ng1!? is surprisingly strong. The Mason Gambit-style 4...Nc6 5.Qh5+ Ke7 doesn't work after 6.d3, and the Bishop's Gambit-style 4...Bc5 doesn't work after 5.d4.
So Black's best is 4...Nf6 as per the King's Knight Gambit, and then 5.g4 follows (5.d3 as per the Fischer Defence has also scored quite well for White).
Now the Quaade approach doesn't work because if 5...Nc6 6.g5 and Black cannot play 6...Ne4. 5...Bc5 6.g5 0-0 7.d4 exd3 8.gxf6 Qxf6 9.Bxd3 looks like an inferior relative of the Muzio/Polerio gambit.
So I think Black's best is probably, perhaps surprisingly, to go into a reversed Allgaier Gambit with 5...h5 6.g5 Ng4.
White's best may well be 7.d4 intending 7...d5 8.f3!?, for after 7.h3 Nxf2 8.Kxf2, after 8...Qxg5 White can't play 9.Nf3 and so Black has time to play ...d5 aiming for ...Bxf5, and after 7.d4 d5 8.h3 Nxf2 9.Kxf2 Bxf5 Black probably isn't far short of full compensation for the sacrificed knight. After 8.f3, which is currently untested, White should be at least a bit better, though play can still get very wild, e.g. 8...Bb4+ 9.c3 Bd6 10.Qe2+ Kd7, when White must avoid the tempting 11.Qe6+? Kc6.
Also, a reversed Rosentreter Gambit (5...d5 6.g5 Bxf5 7.gxf6 Qxf6) may be playable. The only games I can find in my database with 5...d5 instead saw Black meet 6.g5 with 6...Ng8?!, which is hardly in the spirit of the gambit.
After 4.Ne5, although I'm not so well versed in the theory, it seems from a quick look that the line 4...Nf6 5.Be2 Be7 6.Bh5+ Kf8 may be Black's best chance, though I think White must be better in the lines following 7.Nc3, although it requires careful handling in deciding the right moment to hop the knight into f7.