I think the J is a good card for you because it is unlikely that he bets the K-turn with a Jack. You are never calling with a worse hand and it seems like a bad spot to turn his 2nd pair into a bluff.
A bet on the river looks a lot like a J that was afraid to play a big pot but now made trips on the river to get value from a K. If you bet, I think you have to fold to a raise.
But I like c/c more as he probably valuebets a good King anyways and tries to bluff you off your hand a decent amount. If you lose I think its more likely to a full house than to a J.