I think it's hard to avoid getting stacked in this situation.Villain will have AA,JJ,KJs sometimes in that spot also will have some AK,QTs,KQ,KTs,AT combos.Not sure how often he will bet the turn with weaker Kx hands but I assume average players does bet with KQ or draws often enough.Alternatively you can go for bet,ch/call,ch/call line.
I really don't like the turn action! Are we checking to induce? In which case we should just be calling to keep his bluffs in and let him stack off river! I think it is very hard to avoid getting stacked no matter what but as played we are only ever getting it in behind really! The only way this line makes sense is if we are up against a complete spew monkey!
I think I prefer to carry on betting on the turn. Doesn't have to be big but I still expect villain to continue with Jx and maybe TT 99 if he's sticky.
Either way if we choose to check to him on the river and he shoves on a blank it's still tough to get away.