Howdy,
Betting flop seems fine to me, they may have worse pp, I wouldn't like to give up hand instantly and we don't want to really see overcards on the turn. They both are tightish, so their ranges are pretty strong, but still - we have a hand that may be the best at the moment, but there are hell of a lot bad cards out there.
I think betting turn is ok, but we have to understand it's thinnish vbet against a nit as overcards usually are going to fold even if he called flop with them, so mostly we are betting against worse pockets, but it doesn't mean he is going to call with them and he may just got tricky with JJ+/77/22 because flop was rather dry so he let you bet out your guts.
Checking back turn is an option, but it may turn out ugly if spade/overcard comes and he bets big because with low pp he should be scared from overcards so I expect him to bet such a hand already on the flop to get rid of your broadways and avoid bad spots while being OOP.
Overall I'd rather just b/f turn and check back river. I'd fold to his shove as I don't believe nit is going to shove worse here.
Good luck!