I tried that but ultimately you lose the queen and can't stop the pawn queening. The following most likely would be Re4+, then if you move f2 it's check, f4 you loss your queen and black queens

The two key moves are Bc3 & Ke2

I did variations where the room doesn't take the queen, you just attack the rook with the king

Sorry last one should be Ne2#

Got a modification on it. I think it works...

Bc6, Rb1, Ke2, Rxh1, Bg2+,Kxg2,Nf4+,Kg1,Ke1,Pg2,Ke2#

? I dunno if my logic is right, but pretty sure that might be it, been fiddling moves for a while...

Bc6, Rb1, Ke2, RxH1, BxH1, Pg2, Nf4, Kg3/h4, Nxg2. That's as far as I've gotten