I think the river is dependent on how much weight you give to him having KQ some of the time. Given the read, and the donk inbetween, I think you can make the argument that he c/rs the turn some reasonable portion of the time as bluff/value sort of hand -- sometimes best against you, sometimes gets there, and sometimes get's you to fold better.

Let me add another, maybe more fundamental (dumber):

Checking through and giving a free card in a 13 bet pot is a disaster, relative to the smaller price of wasting multiple bets when behind, right? So how little equity would we need against ranges here (ballpark OK or show me how to do the math) before it becomes check?

Or is the point that getting 3! off our hand by aggressive opponents is a disaster too?

OK, this kind of answers my question but not really in a way that gives me theoretical insight. HU against UTG we'd probably want to play near GTO, but against UTG+1 we'd want to play exploitatively. Against both, it's gotta be better to be exploitative, right?

So I'm not understanding what you're solving for.

Bet flop, fold flop. Huge mistake people make is peeling on flop here I don't really think it's close.

If you cap and check flop, your hand is completely face up. You allow Villains to bluff with a wide range of hands including 87s, QTs/KQs/KTs, Axs esp if turn brings fdraw etc and they may not check raise some of these types of hands on flop given your strength.

Also a lot of people won't xr A-KJ on flop either so its a bit of a free card play as well.

We don't want to get bluffed?