I prefer to have 1/3-1/2 pot size bet fold with 2nd nuts here.
XC maybe questionable cause I do not know the size of V's river bet. V may probably hero call with set Q, set J. We block Ks and As and only afraid of Ad here. V may only raise here with AdA or AdQ.

I agree with tomono. For pot control purpose you want I'm betting 1/2, your already not giving him enough odds to call with a half pot size bet if hes calling for a flush draw.

I'm probably XC the river depending on the size of the bet. But without any history of the player its hard to tell.

So you have to try to figure out whats his bluffing range here based on history, and then analyze after that point. Obviously in that run out hes only bluffing or has the Nuts. Most TAG players aren't bluffing here either, if you check and he bets half the pot or less, then he might be value betting you with the nuts.

The other options is that if you are willing to call a half pot size bet on the river, then why not take the lead and make a small value bet for him to call with 3rd nuts.

If he re-raises then its an easy fold, and you just gotta let him bluff you there.

What bet are u talking about guys, if we check call turn?
What's the reason for your check Degrees? There are a lot of hands that gonna call ur bet, but they gonna Check behind all the time. Like AdQ or AdJ, or some 2 pairs. I def prefer betting, but probably u had some good reasons to check and I would like to know them)))
As Played I can see only 1 option - check calling reasonable size bet OTR.

Betting the river would probably cause more harm for your overall range and does not make much sense imho. Just x/c. If he bets too big - you are protected by the nuts in your range, so you can consider x/f. Not sure where the treshold for "too big" is though.

Alternative would be to bet so small that he had to call with his baby-flushes - but what do we want to bluff with? And if he raises over our small bet - we are in the same bad situation, so betting because we being scared is no solution. :D

What about bet/folding turn? Villain could bet his entire range OTR as we rarely have flushes in our turn x/calling range.