TPTK on paired board vs LAG

[Mark1808]

Live $5 $10, villian plays 90% of pots, raises often, bluffs often and has just gone on tilt after calling off a $500 river bet with one pair. We both have $1,500 plus.

Several limpers to button who makes it $55, I call in sb with AQ, villian calls right after and 5 players see the flop of

QJJ rainbow

I bet $200, villian flat calls

Turn A and 2 diamonds giving me top 2 on a paired and draw heavy board.

I bet $250 and Villian raises $340 (pot is now $1,515).

I have seen villian raise the turn with a flush draw and raise with TP weak kicker. I know that he sees me as a tight solid player who is not playing big pots without big hands. At this point I feel that most of the time he has a J here or K10 but at least 25% of the time he has a Q, A or draw here and I am getting over 4 - 1. I have also seen this guy call flop bets with air in position hoping to take down pots on the turn.

My first intinct was to muck but then I began to really think that if I shoved my remaing $1,000 ($600 or so raise) he might very well lay down a weak J given his estimation of my range there and surely lay down the same hand AQ. How should I respond to his raise?