[spamuell]

You might as well bet. He's probably got a worse hand whether it's a pair or K-high and has decided either he's going to pay off one bet on either the turn or river but if you bet both then fold the river, or that he's going to call down. As you don't know which, just bet and hope it's the latter so you can collect 2 bets instead of one.

I don't think it's worth checking the turn to induce a river call as I doubt you're going to collect a call on the river from a hand that would have folded if you'd bet the turn very often at all because he's going to be suspicious and expect you to bet the turn a lot of the time so he's unlikely to just check-fold it.