Chess Online

A position like the following would be more than enough to prevent any other solution from being viable because of the threat of Qh2+.

Your board allows an additional mate 3. Bf1+. The rook on h2 blocks that solution.

Dual solutions in the moves up to a checkmate are usually deprecated in endgame problems, but not necessarily dual checkmates.

However, if you want to have only one checkmate, you still don't need the extra rook because you can put a black bishop anywhere on the b1-h7 diagonal with the exceptions of squares d3 and e4, of course, or you can put a knight somewhere like the e5 square. This eliminates the problem of the dual checkmate because the black piece could interpose on the d3 square, making it one move too slow compared to Bb7#.