grunch
Pre I hope you're doing this with at least QQ+/AK/Axs/56s-89s oslt. Flop you have more equity but villain has nut advantage + we block the NFD, I'd default to putting this combo in my checking range and betting without the NFD blocker. I'd also go either 80-100 with my full range or check a good amount and bet more like 200 when I bet.
On the turn I would just rip it in, it's less than a 2x pot overbet, prices out draws, and puts most of his range in a real tough spot.
Shove river yourself probably, you block the nfd and thus a lot of his air range, I think a check lets QQ/KK off the hook while making you pay off JJ. As played snap river he could even be shoving worse, though it is unlikely.