With two overcards and no draw on a paired board, prefer checking back flop or using a small geometric c-bet instead of sizing up.
Checking back is our preferred play here, though betting small at roughly twenty percent of the pot is a viable mix. Sizing up to half-pot puts unnecessary pressure on our own stack when we hold no flush equity.
Note: Sizing up to half-pot is an over-investment with unmade overcards. Solver prefers checking back or using a small 1.1BB sizing on paired boards.
Checking back the turn is standard after our flop bet was called. We have virtually no showdown value, and firing again into SB's sticky calling range without equity turns a playable overcard combo into an overcommitted bluff.
Folding is clear facing SB's lead. Even though SB's small bet offers generous pot odds, king-high cannot beat any value holdings like Tx or pocket pairs, and raising as a bluff accomplishes nothing with no meaningful blockers.