Welcome to the forums! Interesting first hand.
I think you should bet the flop bigger for two reasons. First, you're multiway on a wet flop against splashy players. There are a ton of draws and made hands that can call you. Second, you're 250BBs deep and you have the nuts. You need to start betting big now to try and get stacks in on the river. I'd bet $60, or maybe even make an overbet if some of the callers were particularly splashy. There really are a ton of draws available on this board that can call you.
Same thing on the turn: bet bigger. If you go even a little bigger on both the flop and turn, then a river shove is only about 75% pot. As you've set it up, a shove is an overbet.
As played, I like a bet of around $200 on the river. You can't be sure that a rivered flush will bet for you; nor can you be sure that a flush will call your check/raise even if you get a bet. But it will be hard for a rivered flush to fold to a two-thirds pot bet here.
Also, I think you need to reevaluate your assessment that villain is "pretty snug" when he shows up with J
7
after a $15 preflop open from UTG+1.