I don't see why you need to put in $15 with k8 pre. The button should widen your opening and 3-bet range but shouldn't be forcing you to make unprofitable calls. You're never ahead pre, and if the K hits you could easily still be behind, and when you flop a draw it's never to the nuts.
I would shove the flop because I'm not a fan of calling 1/5 of my stack then folding to a non K/8/
. As played shove turn