Aces on double paired board

Probably range betting that flop small, like 1/4 or smaller. You could then feasibly range check many turns too. Checking the flop isn't awful but your hand is slightly more faceup than betting.

I think AA is a nice hand to check here OOP so that you can go for two streets when flop checked. That obviously also gives you the ability to bluff a lot.

flop check is fine/good but turn is a bit awkward, might prefer a check to a bet; going for value on river fine as played

i think qqx flop is pretty neutral, so Hero or solver can check some aaxx and bet some aaxx combos, and as long as your strategy is balanced the exact line isn't gonna be super important - if Hero does bet flop, it is going to be for small sizing

with the at and t8 redraws is also kind of nice, as it means hero can xc a lot of turns - but not having flush redraw is a reason to bet the flop