200NL AA facing minraise on drawheavy flop

Villain is 18.5/7.5/2.3 over 1500 mostly datamined hands, and seems solid from the ones I have played with him.

Hero can't remember his own table image from the session in question, but is likely to have been very aggressive.

Full Tilt Poker
No Limit Holdem Ring game
Blinds: $1/$2
9 players
Converter

Stack sizes:
UTG: $73
UTG+1: $57
MP1: $99.80
Hero: $207.80
MP3: $285.40
CO: $222.70
Button: $218.50
SB: $26.75
BB: $190.40

Pre-flop: (9 players) Hero is MP2 with A A
3 folds, Hero raises to $7, MP3 folds, CO calls, Button folds, SB calls, BB folds.

Flop: 3 5 6 ($23, 3 players)
SB checks, Hero bets $19, CO raises to $38, SB folds, Our brave Hero 3-bets to...?

Edit: I really need to learn to bet the full size of the pot on these flops, right?

No way I 3 bet this flop.

im thinking call and bet 2/3 pot on turn with intention of folding to a shove. But i haven't played full table in a while.

Is a 3-bet here bad? Can you post your thought process?

This is my reasoning for a flop 3-bet;
I think I cbet a little too much and because of this I think alot of the time he has a draw here. I'm thinking he's either on the draw or he's got me smoked, and every now and then he's got an overpair. The times he has an overpair there are a lot of scary cards than can fall on the turn, and obviously I will be toast if he hits the draw.

Aww crap, I just realized that if I 3-bet the flop and he smooth calls, then I'm in trouble on the turn no matter what. I'll have half my stack in the pot, and I'll have to either give a free card to the draw or push into it. The push will fold out the draw, perhaps get called by another overpair (which is not too likely because there was no preflop re-raise) and obviously be called by a flopped set or straight. So my flop 3-bet might give me a little value against the draw and overpair, but lose me my stack against a set or straight.

Am I close?