No reads on this reg.
I decided to check the flop, villain has missed very often and should take stabs. I would also expect him to bet all his pairs on the flop for protection. Same for FD's.
Typical hand that people check back is A2, but I block that.
Then the 7 pairs and he starts to overbet. Do you call down?
I think yes, if he ever has a 7 then it's just a very few combos. I don't know what our range looks like for calling the overbet. When we check twice we certainly do rep weakness. I just don't know if villain would find the motivation to overbet bluff the river here. What do you think?

PokerStars - $2 NL - Holdem - 3 players
Hand converted by PokerTracker 4

BB: 100 BB (VPIP: 43.96, PFR: 26.37, 3Bet Preflop: 11.36, Hands: 91)
BTN: 106.75 BB (VPIP: 20.27, PFR: 14.86, 3Bet Preflop: 14.81, Hands: 75)
Hero (SB): 100.5 BB

Hero posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has A A

BTN raises to 3 BB, Hero raises to 9 BB, fold, BTN calls 6 BB

Flop: (19 BB, 2 players) 7 2 8
Hero checks, BTN checks

Turn: (19 BB, 2 players) 7
Hero checks, BTN bets 22.5 BB, Hero calls 22.5 BB

River: (64 BB, 2 players) 4
Hero checks, BTN bets 75.25 BB and is all-in, Hero ????

How frequently are you cbetting this flop with non-made hands?

It seems like a good flop to cbet in a 3bet pot since it shouldn't smash anyone's range here, so cutting out the top of your range by checking AA seems bad for your range overall.

As played ott the 7 doesn't change things much and since villain elected not to bet the flop I would assume leading is going to be better to extract value from made hands trying to reach showdown as opposed to checking to induce bluffs.

I don't know what to tell you to do now otr since I don't think I would have gotten there in this way. It's going to depend on your population reads if you can overfold in this spot so much you fold pretty much the tippy top of your range.

This seems like the perfect flop to bet for you entire range.

If you 3-bet light you can have 2-pair, gutshots, straight draws, flush draws as well as overpairs, AK, etc.

I don't think of the As as blocking calling hands. Lots of hands can call.

I don't know exactly what you range is here, but in general you should have a perfect mix of bluffs and made hands if you bet this flop with almost your entire range.

Given the positions I'm just barreling off starting on flop. We need AA in our range to balance our bluffs on that texture. If we check somewhere I'd probably check this particular turn.

As played, I always talk myself into calling river because nut blockers, under repped hand etc. but I feel most players are weighted heavily in value for these overbet spots. Probably a fold but I'm usually sigh calling it off given how we chose to play the flop.

I prefer just cbetting the flop tbh but the intention of x/c isn't too bad and your logic makes sense but there's still more value in cbetting this flop imo.
I'm not a fan of checking the turn here after checking the flop at all. After flop went x/x you really needed to start going for value and since villain checked back flop he likely doesn't have a FD so probably doesn't have much equity at all but still has a decent SD range after checking back flop.
Villain betting pot on this turn is very odd, especially since you say he's a reg and not a fish. Already I'm thinking he's likely a fish reg. If he checks back flop with the FD then he could do this with a FD. He could also do this with a GS (you'd think T9 bets flop) and GS probably should too mind.
Personally I think he has **** all and reads you as weak with your range capped to AK with that check on the turn. As you've really underplayed your hand and somehow have the top of your range you HAVE to call it off. Villain likely thinks you have AK/AQ etc that will fold to a river shove. He also knows you never have a flush here after you checked flop and turn so it's somewhat of a good bluff card for him.
Call. Sometimes he turns up with a 7 but do you really think he's betting pot with such a strong hand after your range is perceived to be capped at AK high? Nope.

Call but don't play the hand like that again, dude!

I think flop and turn are both debatable /questionable. As played I call river pretty quickly your hands looks like AK AQ imo.

snap (vs. reg)