1 is you lose your bishop to the white pawn next. Then the King has more mobility to get away from your queen/rook
And, you still haven't broken the stalemate of the white rook keeping your pawn from promotion. White just never moves the rook and black can't move his rook and that leaves a knight and two pawns against a king and two pawns... With a white rook covering 7, while the black rook just stares on. The bishop sacrifice at that moment gives black a checkmate in 3 (you're right not 2).
How do you see it playing out if black takes the pawn?
(There are a couple things white could do here, including taking the pawn at g6+, but nothing that really puts white ahead, because black is eventually looking to promote the pawn for discovered check. White can't take the pawn with his rook before it's promoted because he loses the rook. He can't take the promoted pawn either because he needs to get out of check from the black rook. Uncovering the king by taking the bishop with his pawn would work to white's disadvantage against a rook-queen attack.)
Then white's only move would be:
Nf2
Then black could either take the white rook with the new queen or try to maneuver into mate with the rook and queen.
I don't see a huge difference - black loses - or at least risks losing - the bishop either way, and also is able to promote the pawn either way. The white rook can't take the new queen because he is in check from the black rook.
My solution opens up the king's defenses a little and gains black a pawn
That is true. White doesn't have to play Rxb4 after Bb4 either.
Once white stops the discovered check with Ng2, black can either capture the rook at b7 or try for mate. White can't move to the king to g1 or h1 because the new queen is covering that row.
In either case, you are moving the bishop out of the way for a discovered check. My question is wouldn't it be better to get a pawn out of it?
That is true. White doesn’t have to play Rxb4 after Bb4 either.
It doesn't matter whether white takes the bishop or not, in any case black can promote without being taken by white's rook, for example 1...Bb4 2. Ng2 b1=Q, and white can0t take the new queen.
Once white stops the discovered check with Ng2, black can either capture the rook at b7 or try for mate.
I don't think I follow you. Do you mean after 1...Bxf4 2. Ng2? How can black capture the rook? If the pawn promotes, the rook simply takes the new queen (1...Bxf4 2. Ng2 b1=Q 3. Rxb1). And trying for mate without promoting the b2 pawn seems unnecessarily complicated to me (I mean, the position is -5 in favour of black according to the engine, so probably there is some path to mate, but it isn't obvious)
White can’t move to the king to g1 or h1 because the new queen is covering that row.
White moves the king before the pawn promotes: 1...Bxf4 2. Kh1. And again, if you promote, the white rook simply takes the new queen 1...Bxf4 2. Kh1 b1=Q+ 3. Rxb1
I see. You are planning on moving the king before the discovered check. That was not mentioned in the original solution, and also changes the play for:
... Bb4
If we use your solution we can't use discovered check for the same reason, because:
My question was: what would happen if black did not move his bishop to b4, but rather took the pawn at f4. So far, the answers I keep getting begin with b4
which in all cases stop the threat of 2... b1=Q because now 3. Rxb1 is available
The move Bb4 has two functions: clearing the way for a possible discovered check in the second rank, and shielding the promotion square from white rook. On the other hand, the move Bxf4 only clears the way for the discovered check, but doesn't protect b1, so promotion is no longer possible.