I'm leaning towards a flop bet. You're deep enough that you won't be committed if he shoves on you, and while he certainly won't fold JJ, I think your main target for bluffing here should be AJ, AQ and AK. I'm not sure what he'd do with AK, of course. Some people call with those hands. Then again, your basic approach might as well be "I bluff, and if he calls or raises, I'm done with the hand." It takes some serious cojones to pull a bluffraise when you're 130bb deep. He might go ahead and pop you with a flushdraw, but the majority of the flushdraws he can have beat you anyway, so no foul there.
As played, I like checking behind on the turn. I think big aces constitute a large portion of his range, and the rest of it is dead scared of the ace that just came off, so unless you somehow manage to induce a bluffshove from KQ by betting the scarecard, I think checking is just fine. You're not primarily checking behind to induce a river bluff, though, which I think is important to keep in mind. I don't expect you to be ahead 50% of the time if he bets the river (although you'll probably have odds
Once he checks the river again, his range doesn't change much except discounting AK quite a bit since I'd expect him to be more likely than not to bet that on the river once you check behind on the turn. The river bet-sizing is a tough decision. You can bet small and try to get some value out of JJ or JT (if that's ever in his range) hoping to catch a bluff, or you can bet big to make him make a hero call. Or you can bet small hoping to induce a bluff-raise.
All things considered, I like a smallish bet.