After you have 3 bet, i would never ever check this flop tbh unless going for a check raise. So pot control is out of the question now (imo of course). And as played if you check he is only going to bet with a polarised range (because he may be suspecting a check raise), so you dont really gain any info at all based on whatever he decides to do. For example he may take a free card with a draw, or showdown value he would want to keep the pot smallish. But if he has big hand he would build the pot, and if he has nothing hes gonna use position advantadge to try and win.
Ive been thinking about this hand and AJ in general recently. Its kind of a sick spot being oop to a tough player. I would much rather 3 bet a fish with AJ. A good player is not going to really peel with a range that you are crushing here. For example he might fold Ace x, but peel with 78s and small pairs, he may 4 bet with a polarised range of very strong or really weak hands. And AJs is a hand where closing the action, you really wanna see a flop with and dont wanna have to fold to a 4 bet.