50NL 3bet pot IP facing c/r

Villain is a goodish reg. Unfortunately lost my current DB today when Postgres messed up (haven't been able to reimport all HHs yet), so only had around 150 hands on him (even though I've played thousands with him). I forget what his actual stats are, but at the time I had him at 31/28 with a 60% Fold to 3bet.

Maybe this is fancy play syndrome, but I think his flop C/R is pretty FOS. Basically because I'd expect he is 4betting preflop with most hands he's wanting to get it in with on this flop. That is, I'd expect he is 4bet/calling with all his strong overpairs (TT+), rather than flatting against me preflop OOP, as we've tangled enough in the past he knows I'm 3bet bluffing here, and capable of 5bet shoving light.

So on a board where he knows I'm cbetting extremely often, I think his flop C/r is just an attempt to get all my bluffs to fold.

Do I just fold anyways? Do I 3bet/fold the flop small? 3bet shove? Can I flat, intending to bluff future streets, since it will look very strong, since flatting isn't a line really indicative of air in this instance?

Essentially, I just feel his line is kinda FOS, and that he likely has a weak range for doing this, and don't know how I can exploit this.

$0.50 NL (6 max) - Holdem - 5 players

Villain(CO): $50.75
Hero (BTN): $53.59
(SB): $25.85
(BB): $64.74
(UTG): $50.50

(SB) posts SB $0.25, (BB) posts BB $0.50

Pre Flop: (pot: $0.75) Hero has A 3

fold, Villain raises to $1.25, Hero raises to $4.50, fold, fold, Villain calls $3.25

Flop: ($9.75, 2 players) 5 5 7
Villain checks, Hero bets $4.00, Villain raises to $12.00, Hero ???


Also, thoughts on whether preflop should be a call over 3bet? And whether we should cbet or check back flop?

This spot is so read /opponent/ level dependent I think.

I've seen tons of players peel pre here with like 88-TT type stuff, hoping to get it in on a good flop. This is especially true if villain thinks you might play back or get in AK/AQ or something here.

decent regs usually don't flat 3bets pre with weak hands oop, so your $4 into $10 should never see any folds here and you'll need to barrel. Probably not a bad idea to check the flop and fight better battles. Now that you did bet, which is fine, I would just fold and jam stuff like 89s or AK instead.

There is absolutely no reason for you to jam this hand, when you have sooo much other stuff in your range to jam with with much better equity. Fold.