As played, check calling is disastrous imo. We lose to three combos at most and beat a ton that may call, so we can comfortably value bet here. I think the only question is the sizing, as I'd expect nearly everything to fold va a jam and not bet when checked to.
I don't think we get any more value from anything that loses to AK. I kind of want to c/f river because I really hate his turn call. This is why I don't really like betting turn. When we check turn, we rep JJ/QQ and he might try to push us off it. I feel much more comfortable calling down than I do trying to extract value. There's no reason for us to think he's calling JJ or whatever, certainly on the river.

What calls the turn and then calls another bet on river? If this guy is a fish then for sure I'm triple barrelling, not folding ever, but against a nit I think we're in bad shape when he calls the turn.