I'd actually consider betting. Why are we so convinced that we're toast to villain's K? Villain made two virtually automatic bets on the flop and turn, and then slows up at the first sign of hero's aggression. A lot of players would re-raise with a strong K here, and the only K in most players' 3-betting range is a strong K.

I think we're ahead most of the time here, but how often do we expect a call? Obviously if we're beaten, we lose 2 BBs, and if we don't get called, we win 0, and if we're ahead and get called, we win 1 BB. So it all hinges on the probabilities . . . is a 75% chance we're ahead and a 75% chance we get called too generous? Because that would be +EV. I suppose the river bet is marginal.

At the very least, let me say that there is absolutely no question that I'm check/calling this river, if I'm not betting. The decision between the two would be based on reads.