V UTG+1 ($300 eff), old guy
H SB

V limps, hero raises $15 with AT, V calls

Flop AT2 ($33)
Hero bets 12, V raises 69, hero calls

This big raise had me worried about a set, since there don't seem to be any reasonable bluffs. Obviously can't fold top 2 given he could also play A2 like this.

Turn 6 ($171)
Hero checks, V bets 100, hero calls

Not sure whether I should have continued betting into him here.

River 7 ($371)
Hero checks, villain checks

Bigger pre, bigger otf, crai turn, ap lead shove river

Good shout. The raise really threw me off and I messed up the later streets as a result.

I usually do $12+$3 per limper -- should I be sizing up here because I'm in SB OOP and with only BB/limper left to act?

I bet small because (in my mind) the board was pretty dry and I blocked his continuing range, although I guess the only hands that call my cbet are AX or TX which will also call for ~$20-$25, and maybe broadways with a gutshot which I should be charging to see the turn. The only hands I get extra value from that may fold to a larger sizer are smaller pocket pairs.

To be a little more specific:
*Make it a little bigger pre because you are out of position so just taking it down rather than playing it out is less of a travesty.
*You hit top two and this guy is an unknown in the live low stakes, make it at least 2/3 pot because he will call with worse. In fact, many players will interpret your sizing here on the flop as weak and some will attack. If you accidentally induced a bluff raise that's actually kind of nice. What a size nit exactly pot sized raise by villain...
*Turn is such a brick, unless you have a read that this guy is a complete nit there are only 2 combos out there that beat you (I'd be shocked if he has 22 here ever) and we should heavily discount the 1 combo of AA so 1.2 combos beat you. I like 67o's suggestion to shove turn. He is most likely spewing AXs.

your hand is strong enough on this board that its a cooler if he has you, for 300 bucks.

It's a lot easier to get stacks in by the river using a bigger size otf, then bombing turn. Villain most likely has a decent amount of Ax otf, and is probably folding his pp's to any sizing, so I'm really on focused on targeting Ax.

Flop: I like the 1/3 bet on flop. The flop is dry and you have it smashed but because there are two aces gone its less likely he has an ace. By betting smaller you get called by more and you can balance that with and make your bluffs cheaper with KJ and QJ. With 6 combos of sets that beat you and 6 combos of A2 you beat plus villain could be bluffing and could over valuing AJ or A9 I would call raise.
Turn: Turn you could x/c then shove most rivers because most low limit plays call more than they bet or you can x/r if you think villain wont bet or call if river makes the board scary.
River: I would shove don't want to risk a x/x when villain has Ax hand

Flop oop I am betting 2/3p. Bigger bets are less likely to get raised. I'd do this with tpgk, 2p, sets other than top set, draws. I check everything else oop.