If I am going to c/r I would do it on the flop. Getting hands like AK/AQ to fold would be good.
Because the turn is a K I wouldn't c/r. I think Villain is less likely to have a J than Hero. But AJ or KJs/QJs aren't out of the question.
The river can't be better for you so you can't c/f. Villain's call on the turn would be far worse if it wasn't a K. But aside from AK/AA/KQs we could be behind though I think TT would be a longshot. If you are concerned that Villain could have a nut flush or K
Q
then betting half pot would probably only get called (because villain would have to worry that Hero has JT/KJ). And if you bet half pot and villain shoves in theory you could fold because no way Villain is shoving a hand worse than a flush (FTR I don't play online so ranges could be different than live and also in real life I never fold here so I would shove river mostly but I don't really think we will be called by worse except maybe by AJ. Check/calling the river is also an option because it will look like we are scared of the river. But really Villain is never floating so I doubt they will bluff. Jx would check back a lot. Boats or big flushes won't so its better if we bet).
edit: If you have a J then there are a lot fewer combos of AJ/KJs/QJs/JTs that Villain can have. Also Villain will be folding some AJ/QJs/JTs UTG (at least they would live).