I wonder if you could shave off some more bits by considering the specific movement restrictions of some pieces:
Bishops can only ever occupy squares of their respective colour, so you'd only have to encode ~32 possible positions instead of 64.
Pawns (before promotion) can only move straight forward and diagonally forward, which makes their range of valid positions a sort of upside-down triangle shape, with the "tip" of the triangle at the respective pawn's starting position. (e.g. it's impossible to move the pawn from A2 to H2 - or even to H8 - without a promotion)
Haven't made the exact calculations, but it might be possible to encode both positions with 5 bits each instead of 6 bits.
Bishops can only ever occupy squares of their respective colour, so you'd only have to encode ~32 possible positions instead of 64.
Pawns (before promotion) can only move straight forward and diagonally forward, which makes their range of valid positions a sort of upside-down triangle shape, with the "tip" of the triangle at the respective pawn's starting position. (e.g. it's impossible to move the pawn from A2 to H2 - or even to H8 - without a promotion)
Haven't made the exact calculations, but it might be possible to encode both positions with 5 bits each instead of 6 bits.