Villain plays snug, but has a target on their forehead.

I feel like all my choices are bad, but check-folding is least worst. If I were to c-bet I would bet something like 1/3 to 1/2 the pot.

Winning Poker Network (Yatahay) - $0.25 NL (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

MP: 159.08 BB (VPIP: 31.64, PFR: 24.29, 3Bet Preflop: 9.38, Hands: 182)
CO: 101.4 BB (VPIP: 28.05, PFR: 18.29, 3Bet Preflop: 6.45, Hands: 86)
BTN: 244.36 BB (VPIP: 30.77, PFR: 22.12, 3Bet Preflop: 11.76, Hands: 107)
Hero (SB): 137.44 BB
BB: 100 BB (VPIP: 24.65, PFR: 20.00, 3Bet Preflop: 12.09, Hands: 223)
UTG: 101.56 BB (VPIP: 18.18, PFR: 10.80, 3Bet Preflop: 1.45, Hands: 182)

Hero posts SB 0.4 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.4 BB) Hero has K K

UTG calls 1 BB, fold, fold, fold, Hero raises to 4 BB, fold, UTG calls 3 BB

Flop: (9 BB, 2 players) 3 A 4
Hero checks, UTG bets 4.28 BB, fold,

We should cbet 2/3 - 3/4 pot for value. We can still get value from worse pairs. Villain can sometimes have Ax here, but the times that we win (when our hand is good and when villain folds) more than compensate the times we lost. And check/calling vs a passive fish is bad because he will only bet with Ax here and will check back all his airs/worse pairs for showdown. Although check/folding is better than check/calling, we still lose tons of value by not cbetting.