It depends a lot on what you think of the EP raiser. AQ isn’t going to match up very well against an EP’s range.
Pre-flop, I don’t mind seeing a flop because you have a nice drawing hand and position. I wouldn’t 3-bet it, I don’t think, unless you know he’s coming in light a lot. If he were in one of the MP spots with a caller behind, I would 3-bet and squeeze. I tend to give EP raisers a lot if credit. AQ isn’t going to do very well if he wants to put all the money in pre-flop.
On the flop, I would not raise. I would flat call. The flop is AJ9. So really, if the raiser has an A, 90% of the time it’s better than yours. Almost never is it worse. The only semi-legit A high hand you beat is AT. That tends to be at the bottom of the raising range in EP. If he has KK, QQ or TT, you want him to stay in the hand. He’s drawing to 2 outs. It’s worth waiting to see if you can get another bet.
When the J comes on the turn, if he checks, I would check behind. I’m looking to get to showdown with a hand that has some value. If he bets I would fold here a lot. When he fires twice on this board I’m usually behind. If he checks he may be drawing, but there is very little he could be drawing to. On the turn I’m managing the pot and under-repping my hand. Right now we’re in one of the classic way ahead, way behind sorts of situations. Without a read that he will stack off with 2nd pair, we don’t particularly want to build a big pot and we’re in a great position to give cards. I’m checking to induce a bet from a hand that missed. Again, my hand has showdown value. The most likely outcome if I bet is that he folds. If he doesn’t fold, we’re usually crushed. I would prefer to get some value out of the hand on the river from hands like KK and QQ that might stick out a bet or call a 1/2 pot bet. I’m going to call most reasonably sized bets the river.
As played, unless you have a great read I would call the bet. It’s a good price, which is always suspicious, but you do beat some things that would bet. I’m a little surprised he showed up with KJ and that he called a raise with it. I wouldn’t feel so bad that you lost money to the guy. When he gets so much money in light, OOP, he’s bound to give it back at some point.