KK - fd on monotone flop

Can't find a FTP converter that works for this forum and don't wanna kill all of your eyes so I'll type this up. Villain is a TAG regular, he's 16/14 from what I have on him (600 hands). Most of those hands have been datamined, but I have stacked him once before with a set vs an overpair, no other big pots so far.

Effective stacks $22
I'm UTG+1 with KK and raise to $1. Folds to villain in BB who flat calls.

Flop 85J
Villain checks, I check behind

Mixing up my play, I'm probably not getting much from this villain unless he has the A, otherwise I'm probably behind.

Turn 85J [ 3 ]
Villain bets out $1.50 into the $2.25 pot, I flat call.

River 85J 3 [ A ]
Villain bets $3.75 into ~$5.25, I flat call.

Any reason to raise this turn? River I'm looking to get to showdown cheap, even though it was the worst card in the deck.

I think folding this river seems optimal because villain can logically put us on the A (if he doesn't have it himself), so if I hit my ace on the river, I'm gonna call, there's no point in betting a worse hand right? I think river is spew, but could be shown otherwise.

Also, if we're raising turn, how much are we raising to? $6 sounds about right IMO.

I definetely raise the turn, I definetely bet the flop (Draws/combo draws are going to look to raise, which is money for us). As played I fold the river. Looks like an A of diamonds, could be two pair, you'r only beating a bluff.