Can Black save this Knight vs Bishop endgame with a pawn down? After playing horribly in the opening, my opponent played a couple of suboptimal moves which suddenly allowed me to exchange some pieces and get the following unclear position.

I thought about giving the h pawn away, giving White two potentially
useless h pawns if he captures gxh4. However, I thought that making 
use of it would require trading my g pawn for the White f pawn, which
seems to be impossible.

[White "Villain (2200)"]
[Black "Hero (2000)"]
[fen "2B3k1/5pp1/4pn2/8/3P3p/4P1PP/5P1K/8 b - - 0 1"]

1... Ne4 2.Kg2 hxg3 3.fxg3 *


The position after fxg3 was when I needed to formulate a plan. I thought of several options, all of which are more or less passive defenses, as I think moving my pawns would only weaken them and gain nothing in any position.

First idea: Leave my king on g8, and sacrifice the knight to get a fortress. 
More precisely, force White to move his pawn to g6 and trade the 
knight and f7 and e6 pawns for White's other pawns  (king, pawn g6, 
light-squared bishop vs. king g8, pawn g7). 

[Event "Draw"]
[fen "6k1/6p1/3K2P1/8/4B3/8/8/8 b - - 0 1"]


However, I didn't see any
way to force this: if White can keep any pawn not in g6, he wins if I sacrifice the knight.

Second idea: Leave my king to protect the g pawn. This forces White to exchange 
the e pawns. Then I can use my king to stop the passed d pawn and my 
knight to stop the kingside pawns. However, from what I calculated (and analyzed afterwards), it seems it is not sufficient. This is just an example 
of what I think could happen; I haven't checked every move thoroughly, but the general idea is still the same: Black can block 
White's d pawn, but White can always force a win due to Black 
knight's limited mobility and zugzwang.

[White "Villain (2200)"]
[Black "Hero (2000)"]
[fen "2B3k1/5pp1/4p3/8/3Pn3/4P1PP/6K1/8 b - - 0 3"]

3... Nd6 4.Ba6 Kf8 5.Bd3 Kg8 6.Kf3 Kf8 7.e4 Kg8 8.d5 exd5 ( 8...Kf8 9.e5 ) 9.exd5 Kf8 10.Ke3 Kg8 11.Kd4 Kf8 12.Kc5 Ke7 13.Kc6 g6 ( 13...f6 14.Kc7 Ne8+ 15.Kc6 ) 14.Kc7 Ne8+ 15.Kc6 f5 16.g4 fxg4 17.hxg4 g5 18.Bg6 Nf6 19.d6+ Kd8 20.Bf5 Ng8 21.Kd5 Nf6+ 22.Ke6 Nd7 23.Kf7 Ne5+ 24.Kf6 Nc4 25.d7 1-0


During the game I remembered that KBPvKN with the advanced pawn can be a win in some positions. However, I didn't remember when that happens, and I also couldn't see
any way to force the exchange of all kingside pawns before allowing White to advance his pawn this far. In post mortem, I was able to blitz a win against the opponent in a position similar to this (without the g pawns), even though he claimed it is an easy draw.

Third idea: Use my king to stop the d pawn, exchange my e pawn to White's d or e pawn and force him to defend the isolated pawn so he can't make progress. This is what I decided to do in the game, making by endgame skills look ridiculous.

[White "Villain (2200)"]
[Black "Hero (2000)"]
[fen "2B3k1/5pp1/4p3/8/3Pn3/4P1PP/6K1/8 b - - 0 3"]

3... Kf8 4.Ba6 Ke7 5.Bd3 Nf6 6.Kf3 Kd6 ( 6...e5 7.dxe5 Nd7 8.Kf4 Ke6 9.Bc4+ ) 7.Bc4 Nh7 8.h4 Nf6 9.g4 Nd7 10.g5 f6 11.Kg4 Nb6 12.Bb3 e5 13.gxf6 gxf6 14.h5 Ke7 15.h6 Kf8 16.Kf5 1-0


Something like 6...e5 was my idea but I couldn't make it work in any position. For example, after 6...Kd6 7.Bc4, I realized the f7 pawn would hang. I saw no other way to make my position any better: any pawn move would weaken them and any king move would give White more space. Only after 13.gxf6 I realized that my plan of moving king to stop the d pawn allowed White's king to break through easily.

Question: Did I miss something? Was one of these plans drawing after all, or is there a resource I didn't see, or is the game simply lost?
I think it is good to notice the possibility of different fortresses, like the one you mention, and of course the wrong coloured bishop for the h-pawn. But to turn reaching such a fortress into a "plan" is way to early in this position.

I think fxg3 was probably ok. At the first glance it looks ugly that now there are white prospects for a passer on the d-line and on the h-line, but your g and f-pawns can actually deal with the white g and h-pawns. Against g4 and h5 you should play f6 and against h4 and g5 (like in the game) you could have played g6. These setups prevent an h-passer.

But maybe 10.f6 was still ok. Then you certainly should have played 11…fg 12.Kg5 Nf8 which prevents the intrusion of the white king. But even here h5-e4-e5-h6 and Kf6 might just win for white.

An engine will be able to come up with better lines, but my feeling is that


Keeping your pawns compact.
Keeping the white king away.
Preventing a h-passer.


might have earned you a draw. (Coincidentally you violate all these guidelines with the move e5, after which you are surely completely lost.)