On dynamic paired boards, use small sizing to protect your range and check-call when the board texture shifts against your marginal value.
We have a clear range advantage on this Ace-high paired texture, but our specific hand is marginal value. A small bet is preferred to extract value from spade draws and weaker Kings while keeping the pot manageable.
Note: Sizing up to 3BB is slightly too large here; a smaller 1.3BB to 2BB sizing allows us to bet more of our range and keeps Villain's continuing range wider.
The turn is a difficult card. While we improve to two pair, the Jd completes the broadway straight for QT and increases the likelihood Villain is sitting on an Ace or a boat.
Note: Betting here is thin; checking is higher EV to control the pot and realize our equity as a bluff-catcher.
Checking the river is correct. After the turn action, our hand is a pure bluff-catcher, and betting would only be called by hands that have us crushed.