[MainEvent]

I think you have to bet the turn. That leads to a few scenarios.

Scenario 1 - he calls. He either has a pair (possibly Aces) and is resigning himself to calling down or he is on some type of draw (overcards to yours, flush). Either way, the river probably plays the same. You have to check because if he is resigned to calling you down, it is with those pair hands. If he is on a draw and wiffs, he won't call your bet out. Perhaps you get a bet from him on a bluff if you check-call.

Scenario 2(a) - Villain raises. He may be using the scare card against you. You can 3 bet with your flush draw + 2 full house outs. If he has no pair, and is betting the scare card, he is still not drawing that thin. He has 6 outs for his two pair, plus 3 aces. 9 outs. Enough to not want to give him a free card.

Scenario 2(b) - Villain raises. He may have the ace or some overpair to your hand. If he is gutsy/good enough to raise with an ace/overpair, he may be good enough to lay down to a 3 bet. In addition, you currently have those redraws, so I think a 3 bet might be worth it. (Sidenote: I thought about these choices as options but I am fairly confident that I am not a good enough player to make them)