AKo sqz deep.

You are so deep. Eeek.
x/f after betting 6$ on the turn aint an option.
if we think about his range, it's pretty much 10Js and AK. Perhaps 76s. Id expect a raise from a set at least by the turn (although slowplaying from him here wouldn't be bad at all).
So according to that range, x/c seems like the best option for us. If we bet we get his air to fold that he might of turned into a bluff.

Given that hero’s range includes KK, villain might prefer to play a lower set passively when in position. Is TPTK good for 3 streets of value against the best reg on the stakes?

I think check folding can still be an option. After this action I don't think he has that many missed draws left in his range that he would turn into a bluff. You have showed some strength also so I guess he is checking behind with many of his hands.
Bet folding can be an option too. He ain't going to raise you as a bluff, and that way you get to decide what the bet size is going to be, I would go with a smaller sizing.
Check calling is the worst option imo. Because I don't think he is going to bet with a weaker value hand that often and don't see that many bluff hands in his range. I would hate to check and see a jam from villain ♦

It is a tough spot. I played with Snowie to see what it thinks. OTT it would actually prefer checking, and I think it makes sense. We might not want to 3 barrel this hand. If we check the turn and villain checks behind we have a bet OTR. If villain bets we can check call and then check/decide OTR.
On the river it looks like we have several decent options. Snowie would mostly check this hand, but betting 25%pot has similar EV (it would check/bet with 80/20 ratio).

Snowie would check call or bet call this. But we have to keep in mind that the recommended action is against "optimal" villain. Because of the reasons I said above, I think bet/calling would be bad. And not super thrilled about check calling either.

I was alternatively thinking what about a small turn bet.
If he has a set and we bet 4$, it'd keep him more honest as i'd expect to see a raise. This would also keep the pot small on the river and make it easier for us to x/c or bet ourself.

Definitely a tough one, but not a common spot. So lets not get TOO caught up

A very tough spot, indeed. The move that has the most sense to me is betting like 25% of the pot and fold vs a raise. It sucks that our pot odds to call would be just ~20%, but in reality, I don't think we will have it. He probably doesn't have that many busted draws at that point (because of all the action) and even though he is a good reg, I think he would be scared to bluff that deep.