Had an interesting situation in a super small stakes game yesterday, I think this is a very borderline river decision. Tell me what you think, I'm just listing the bets in big blinds to keep it general. And I probably screwed up some math at some point

Game has a lot of obvious players, playing to "hand protect", meaning big but vulnerable hands are announced. Moderate to low skill level, not a lot of tricky play. I also expect calls preflop from very wide ranges (one guy called with Q8 off on a 3bet pre). My image is pretty good so far.

I'm utg+1, effective stack of ~180bb, with KJ, folds to me and I open for 3bb. Get 3 callers. Pot is 12 bb.

Flop is KT7, I lead for 6bb. 3calls, pot is 36bb.

Turn is 4. I lead for 20bb. Again, 3 callers. Pot is 116 bb. My stack is ~151bb.

River is 8 (flush draw misses, J9 comes in, and 69 comes in).

In this spot, would you

1. Bet-fold for value
2. Check-call
3. Check-fold

I was looking at it combinatorically, and having a hard time deciding what was right. By my count there are 40 combos of hands which I beat that I can get value from:

K9, K6, K5, K3, K2 -- I include K2 for sure. Dunno if tens ever call here.

There are 50 combos of hands that beat me:

KQ, K8, K4, J9, 96 -- I'm excluding 2 pairs and sets on the flop because I think they raise. Don't know if pocket 4s or 8s ever continue to hit their sets. If you eliminate 96 from the turn bet, that total combos I lose to drops to 34. All of those hands, except maybe KQ, raise me on the river.

I want to say it's a bet fold for value, but I'm really not sure. Checking could induce bluffs or value bets by worse, so that's the argument for check-calling I suppose. Is this is a check-fold ever?? (This is more of an exercise than anything, I'm nowhere near savvy enough to think this through during a game).

Why the small raise pre?

I'd go 3/4PSb+ on that flop. Pretty wet board. More on turn given we have 3 customers.

Easy smallish bet/fold on that river.

Yeah my early play was questionable. No real justification for not betting more on that flop

My initial reaction after thinking over the hand later was, it's an obvious bet-fold, but after ranging my opponents and seeing how many hands now beat me on the river, I thought that may change things... if there are more possible hands that will call/raise you with better than will call with worse on the river, is a bet-fold profitable? Or is it just less bad than a check-call? Thanks for the reply!

EDIT: Of course, this situation is very range-dependent, just eliminating 96 from their range swings it the other way, where more hands call with worse.