The thing with checking on the turn is that you need to have some balance.
The fact that I don't think you ever check KQ, KK and other good hands which beat paired kings would tell me that if I had a K in my hand I had you beat.
I personally thing you should be betting the turn, as your cbet range on the flop includes a lot more K type hands than your checking range does so it doesn't give away your strength.
A Question for you:
If the river is a dead card say a 2, what do you do to the BTNs bet?