Preflop I 3bet bigger like you said. You're ~140bbs deep. This villain will be calling bigger bets with all kinds of worse hands and probably 4betting a lot of them also where I'm usually happy to get it in preflop. Not only that but you will be more likely to get heads up with him.
I'm checking behind the flop as well, but not because I'm afraid of a set. Of course it's more likely someone hit a set when the pot is multiway, but right now I only have Ahi and a couple backdoor straight draws. If I bet I don't expect too much worse to be calling. Maybe some worse aces, some straight draws or over cards, but I think it's much more likely I'll be called by hands that beat me: 9x or Tx, medium pocket pairs, etc. I can see maybe cbetting here hoping everyone but the fish folds so you can be heads up, but I don't really feel like it's a great idea. I'm much more comfortable checking behind hoping for an A or K.
When the off suit K comes on the turn and the lagfish bets I think I just ship it there. I don't think SB or UTG would be checking their bigger hands. I think the fish would call with most of his Kx hands. He probably calls with his Qx or Jx hands for the gutshot. He might call with some of his paired hands. And if he folds his air, which I think he has lots of that too, I'm happy winning an 85bb pot.
As played on the river I'm not sure, but I probably call too. Looking at his stats makes me feel like he will be bluffing
here a lot with his missed draws and quite possibly air. He likely bets there with most of his kings and maybe worse pairs. I tend not to beleive people running like 90/65 and 5.5 AF over a decent amount of hands. Maybe that's a leak. I dunno lol.
You're getting just over 2.5 to 1 to call so you have to win something like 29% of the time, I think (I might be wrong and would love to be corrected if that is so).
Of course this is just how I look at the hand. I make no claims that anything I say is the right way to look at it lol. I look forward to seeing what others say about it.