I like 1/2 to 2/3 pot. He minraised you to see where he is and by betting out here with a semi-weak bet, he still won't have a good idea where he is.
If he doesn't call or raise the turn, he's not putting money in on the river unless he has a big one.
Keep building the pot and hope he has something like TT, strong draw, or hands he may stubbornly draw very thin with like an overpair.
Unless I know that I have an opponent who perceives any check as weakness and will bet after my check, I always build the pot with weak-looking bets in situations like this hoping to string them along.
The real question is whether you let him draw to eight cards (five ties included) if he does pop it up hard indicating that he probably has AJ.
Curious to know what you did / results.