I would check to control the size of the pot and avoid being check raised, getting to showdown as cheaply as possible would be my plan. Only getting called by better hands the majority of the time on that turn I'd imagine. Interesting he called here with AQ however

If you have a read that villain is passive and shuts down a lot on turns in 3bet pots, I'd maybe bet sometimes to protect our hand

In a 3 bet pot I would expect micro players to be checking overpairs to you on the turn quite often on a paired board in the hope of pot controlling, and paranoid that you can have a good number of 9x in your defending range.

I think if you want to bet for protection then even $2.50 does the job as we're not getting any real value when betting bigger.