I am going to make my response without reading the spoiler.
The fact that he 3bet you a lot today doesn't mean much in and of itself. Good players are going to 3bet a lot and cold call much less frequently. And then him running a little good pre can make it seem like he is 3betting to often. But the hand he has shown down are more important. Is he 3betting you with KJo pure preflop? 76s pure? ATo pure? Or is it more like a normal range? Does he also cold call a lot?
And what have been his postflop tendencies, especially multiway?
Preflop this is a good illustration of why I like smaller preflop sizes. You aren't thrilled to get 3bet.
I think calling or folding is fine to 3bet pre.
Flop is standard.
Turn is standard.
River sucks. He should check flop a fair amount multiway even with Kx.
Just because he is a LAG doesn't mean he doesn't slow down multiway. LAGs like to be bluff, aka they like being tricky. Checking a strong hand is tricky.
I am not sure he takes this line with QQ. I mean, in his mind, your range is mostly capped to not having Kx after checking turn. Unless you just have a monster like JJ or KJ, 77. So he could be going for thin value with QQ I suppose.
I think you actually have a lot of traps in your range. Are you 4 betting JJ? Because you have JJ, 77, and KJs for 7 potential combos, and you don't really need to bet those on the turn. Do you bet them though? Then 3 combos JT, 3 combos QJ, maybe 3 close AJ if you don't bet turn, and maybe some nutflush draws.
I think your hand is very cuspy. Probably better than AJ since his bluffs are going to contain AQ and AT, and he should have a lot of combos of AQo. I don't know if he takes this line with AJ. He could going thin with AA too I guess, although if he does that he should probably do that with QQ also.
I mean, if he had clubs, better chance he bets flop. Even if he had diamonds and a gutshot, I feel like he might be flop more. So you shouldn't be too afraid of flush draws on the turn and you might want to trap some hands like KdTd and KdQd + boats, let him fire off his AQ/AT/QT combinations with hearts and spades start bluffing like a true LAG. Then I think you can safely fold your hand without being too worried about getting exploited.
Okay going to read the spoiler now.