AQ CO vs MP raise - postflop play - board Q J J

I like your line for the most part. I think AQo is a decent candidate to 3b pre in this spot. AQs is a mandatory 3b IMO.

As for the river, seems kind of thin to bet here. He is folding is busted FD's and SD's, calling/raising his Jx or better so your target for a value bet would be something like KQ or sometimes Q10 or maybe TT but it's not a guarantee that he calls with those anyway AND how often is KQ or QT going to check/minraise this flop? Checking back seems right and if you get frisky and want to Value Bet make it small so you can get away if he is trapping the river.

He's line of betting is really strong: pre-flop raise, check-raise on the flop, lead on the turn.
So he's either bluffing or he's there and knows you got something.
I believe you're beaten here. He played the entire hand out of position, showing an incredible strength.
In this spot I'd check-behind the river because it's probably an opportunity is giving you to bluff.

Note: I wouldn't be surprised if he also had AQ.

Well, his line makes me think about a bust FD because of hix x/r sizing (lol) but anyway, you don't value much hands here or maybe you can bet something like 0.25$ if you think he has KQ or even TT.. I joke, there'is no value river imo..