It would help a lot to know the BB's tendencies postflop, but assuming he plays a wider range a hands, I'm also going to assume he is going to be c/r the flop quite a bit w/ pretty much all of his draws, especially if he knows the BTN checks back all of his marginal showdown value hands. Given all that, I think a 7 is the best card for the BB. BTN can't continue w/ any of his air range as BB will have plenty of 7's in his range, and BTN can't really have a 7 if we are assuming he checks back his weaker made hands.

As far as the best card for the BTN, I think most of it depends on what he thinks of BB's flatting vs checkraising range. If BB is more passive w/ draws then As Js 6s are all bad cards, if BTN assumes BB is agro w/ draws they all become good ones. Without this information I think I'd go with an offsuit king being the best card for BTN since it helps the BB's perceived range less often than any card except an ace, and because BTN would check behind an ace on the flop...if BB is perceptive I'm assuming he'll know this. ALSO, a king hurts hands like 88/99/7x/Tx the most because now a ton of BTN's range has a gutter at worst when firing the turn, hands picking up draws being: AJ/AQ/Q9/QJ, plus a ton of his range also picked up a pair.