[SNOWBALL]

Yes, bet the river. Don't worry so much about whether he will call with a worse hand. That's billy's job. Your job is to realize you are almost always best, and to bet your hand.

The turn is interesting, because Billy should have very few outs against you. If he has KQ, , or QT, or JT, he is drawing to 2 outs. If he has QJ, you are drawing to 12 outs. If he has a set of fives, AJ, or AQ, then you are drawing to
7 outs. Overall, you can't fold to a 3 bet on the turn, and it is unlikely you can make 3bb total from him (him calling your turn raise plus paying off on river ui) so calling seems best on 4th street, and the pot is not ginormous, so you don't need to encourage him to fold.