Clear valuebet don't allow villain to xback sdv hands
When check should call villain could vbet AQ potentially and be bluffing with QJs or do something random

I think the answer to your second question, of whether or not to call if we check and he bets, is definitely a yes. AK is simply too strong to consider folding here

But I am not as sure on sizing on the river. Opponent has roughly a pot-sized bet left, correct?

I'm curious what others suggest for sizing on this river. My guess is that no size can be too bad here, but I just don't know which one is optimal.

Thoughts???

A valuebet for sure. He has a lot of missed draws, second pair with missed fd, top pair worse kicker etc.

I wouldn't theres still a decent amount of hands we're getting value from. But i'm not an expert - so maybe a check is better.

Bet bigger on the turn and shove river. AQ/AJ should be a pretty big part of villains range, so we can shove AK+ for value. With worse valuehands I'd mostly check the turn.

Addition to Samu: In these kind of spots I try to go by the following rule: if I call a shove a shove it's better to shove myself. Otherwise villain will check behin AQ and shove TT himself. This obviously changes if there are obvious draws that miss, then it might be better to check to induce bluffs. In this case, with us holding Ks I think i'd rather just shove myself.

I'm jamming here but I am too agro :( I like to capitalize on some fold equity also in these spots.

You are a 55/45 favorite over A10s+, A10o+, AA, 1010 and 44. He might fold AJ some of the time which strengthens his calling range (like 60/40 in his favor) but that also means he is calling less often so you will capitalize on the dead money more often.

It's a slight overbet on the river which isn't terrible but some bet sizing tweaks could make it slightly less than pot.

You have top set in your range, he likely does not. You both should have TT based on prelfop. This is a huge advantage for you. This is by no means a clear cut call w/AK if you shove river. Say you shove AK and he folds AK just some small percentage of the time, this is a HUGE bonus for you. I guess the easy way to describe what I meant by "capitalize on some fold equity" is that some times he is going to fold a split which is great great great result for us.

You are a good player and I respect your posts a lot, so I am assuming your flop and turn bet range include more than just AK+. I see AQss, AJss barreling turn also. These aren't really bluffs but they are getting kind thin from a value standpoint. Say you jam AQ and he still call AK a lot, but he does fold it sometimes. But now he is folding AQ a large % of the time. Again, him folding those splits is a great thing.

I agree it's kinda thin since you are both so deep but this is the microstakes and people play TP poorly sometimes. If you both get to river with A10+, you will have an advantage by shoving river and putting him to the test. It's just not an easy call to make with hands splitting with you or just slightly better than a split.