Post Flop with KK pocket pair

I usually play this 1 cbet and then give up. But betting 1/2 OTT seems ok too. When villain donks OTR, you should fold. Usually donking OTR (at micros) is strong.

Villain called a 3bet so his range has many Ax hands. I think you can still bet once on the flop but if you get called you should be careful. Maybe it is best to just check behind OTF and call one or maybe 2 streets.
Call call donk line is very strong and you should have folded there. He played his hand so "weak" because you have a lot of better Ax hands in your range but since you check twice he fired a thin value bet.

Thanks for the insight into Villain's thinking. I wondered whether he was perhaps concerned that I had pocket aces or AK, which he knew would beat his hand... In the future, I'll proceed with more caution with KK if an ace appears on the flop.

You should cbet 1/2psb on dry board in 3b pot.
Turn bet is too much.
His line is strong and your river call is losing money. With what weaker hand do you think he bet ?

My standard here line would be :
b/f OTF / c/f OTT / b/f OTR.

Checking flop when you are the one who 3bet him PF is standard there, because you have initiative. In fact you could consider him as weak because he's scary of you holding AJ+ (that's what make him a recreational player because when you have hold AQ/AK).

But when he call he has either Ax or a PP that he can't let go. In either case don't bet OTT to control potsize, and you can thin value bet OTR depending on your read.