TPTK vs River raise

How not to play this hand lol. The goal isn't to give the most chips possible without seeing the showdown you know

I'd like to check flop quite often here. You are supposed to check a lot as you're OOP and the flop is dry, the spot is kinda WA/WB and IP will make much more mistakes if you slowplay a little.
If you keep barreling I don't know what remains that you can hope to beat river. You are so face up ! He can have AQ+ and he won't have many AT-

I think this can be played in two ways:
Double barreling flop and turn, check the river.
Check flop, bet turn, evaluate the river.
As I said in a lot of topics, I don't like to triple barrel an unknwon villain with TP. And in this case you don't have the best kicker, so I'd definitely choose to pot control on a given street.

Check flop here usually and if betting, bet smaller. You are likely slightly ahead vs. his TP combos here if his PF calling range includes AQo (Have to assume he 3b AQs in this spot more than likely). I would probably bet/bet/check my AQ combos and triple my AK combos but all with smaller sizing since the range you are targeting is massively weaker than yours when you are betting AK here and less so with AQ.

Betting more polar is the way I would go. Bet the best and the worst here. AK, AQ (especially AQhh and AQcc). I would also add some bluffs with low equity, high potential hands like 63s, 65s and on occasion u can turn a low equity pair into a bluff like 55 or 33 since you heavily block flopped straights and straight draws. I would probably be checking most, if not all sets and straights on this flop. I would consider betting 44 since he will have 22 in his range some of the time, but I still mostly check it.