Each pawn that wants to be promoted either takes: (a) another 'special' piece (knight/rook/bishop/queen), in which case it has already bought enough bit budget to later be promoted; or (b) another pawn, in which case this temporarily saves 1 bit (as the other pawn becomes a space), but then later we need 2 extra bits for the promotion, so we pay 1 bit extra per pawn in total

In the case of (b) there are now fewer pawns that can be promoted, and so worst case, we have to pay a budget of 1 bit per each of 8 promoted pawns.

So I think maximum required bits is only 162 + 8 = 170?