If Black promoted on h1, then Black could have reached the position with only one bishop move, Bc8-h3.
White could have reached the position with only two bishop moves, Bf1-e2-d1.
In order for that to have happened, White must have given up at least three pieces along the a8-h1 diagonal, and promoted pawns to get the pieces back.
White could have given up three pieces and two pawns on the diagonal. given that White has two pawns left, and gave up two, that means that three of the remaining four could have been promoted to replace the lost pieces.
So it looks like the minimum number of bishop moves is one for Black and two for White, for a total of three.