The stipulation is retro so DP rule is switched right away. Next move is certainly 1... bxc5=, so the position is dead. So what alternatives might White have?

R: 1.Bb4-c5+? Alternative 1.Bb4-c3+ enters perpetual check loop with the only escapes being stalemates. So no checkmate is possible (even were 3Rep/5Rep/50M/75M/DP rules removed)

1.Bc5+? bxc5=
1.Bc3+ Kc5+
2.Bd4+? Kxd4=,Kxd6=
2.Bb4+ Kd4

Therefore the prior position was dead already and the last move would have been illegal.

R: 1.Bb4xSc5+? No alternative because bS is checking wK, and must be captured, so again it's impossible by DP rule.

R: 1.Bb4xPRQc5+? not possible because bBd1 is promoted so all 8 pawns, 2 rooks & 1 queens are already visible on the board.

So by elimination it was:
R: 1.Bb4xBc5+! Now the only alternative was 1.Bb4-c3#, so this last move would have been legal.