Avoid over-betting middle pair out of position; check-calling allows you to realize equity and catch bluffs without bloating the pot.
Checking is preferred here to protect our range and keep the pot manageable with a hand that has showdown value but is vulnerable.
Note: Betting second pair here is a slight overplay; checking is higher EV as it protects our checking range and controls the pot size.
Checking is the pure play on this turn. By betting again, we isolate ourselves against better Ace-x hands and invite the raise that eventually forces us off our equity.
Note: Double barreling middle pair into a condensed range is too thin; checking allows us to bluff-catch or realize our equity for free.
Folding is mandatory once we face the raise. Our hand is now a pure bluff-catcher that loses to all of Villain's value range (Ax, 45s, sets) and blocks their natural semi-bluffing combinations.