Given that you were in position, I think flatting the flop could have potentially saved you. The BB's range is pretty wide, and this is not a board texture that he would lead on.

If you do flat, and the BB raises, assuming MP player would still fold, you can narrow his range down to made hands because you have key blockers to the flush, so his hands will more likely consist of stronger top pairs like AK, two pair hands, sets, and the effective nuts (AJ). Also, flatting gives the BB another chance to barrel if he is bluffing, which is unlikely given your holding and the board texture but thats a more player dependent situation.

With the turn being a 6, that is a complete brick. Your opponent can either lead out to you, in which case you can probably only call, raising would only narrow down his hands to even more strength, and depending on the sizing, you could potentially fold. I know that seems really tight but given the Pre-Flop action, the BB could definitely want to close the action with a hand like AJ or KQ, because if he does raise he could face a shove from hands that dominate him.