Check-raising a 2+2er with quads

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I think that it is highly likely that the villain has a kings full of nines full house

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I don't think this is likely as you think. What hand is this player calling preflop and betting the flop with that includes a 9? I think it looks like you have Kx, A3s, or some pocket pair. I don't think this player will be betting this river enough for you to make money on your c/r. If she will fold to your raise, then you only make 0 or 1 bet. If you bet, you make 1 or 3 bets. I think this player will raise your bet often enough for you to make more betting than by checking. This is a perfect opportunity to get aggressive with a nut hand.