While we have two pair, checking the flop is preferred to protect our range on a board that favors the caller's condensed range.
Checking is the preferred play here. While we have two pair, this board texture hits the Button's calling range very hard, and we need to check our strong hands to protect our overall checking range.
Note: Betting here is a mistake; solver prefers a pure check with this hand to protect our range on a board that favors the Button.
Once we bet and face a raise, our two pair is too strong to fold, but the back-and-forth clicking suggests a very narrow, high-variance range battle.
At this SPR, we are effectively playing for stacks. Shoving ensures we realize our equity and denies Villain the chance to realize theirs with draws.