1$ On-tap final table heads up AQs

I would 3bet larger preflop, to 13-14k or so. Giving such pot odds for villain to call from position is not good.

In flop my cbet sizing in such dry board tend to be smaller than half pot, but half pot is absolute maximum sizing. When we get called and Turn it T there is little point in second barreling. I have hard time to see villain folding barely any better hands and getting called by worse is unlikely also. So I would check and evaluate. That way we can at least get him to bluff if he has floated with some air type hand.

When he does shove over our second bet we beat very few hands. I would assume he has hand like 88 or better almost always. But we only have 40k behind to call for pot of 160k so we only need 25% equity. With flush draw and over cards its probably crying call.