What do you do with a persimmon tree?

Yes.

… but, to your question about how to treat bad-tasting persimmons, I can only say I don’t know. How big is the fruit? Did you pick it, or did it fall? Was there frost? ISTR that my family’s American persimmon tree dropped most of its fruit after the first freeze/frost, and we picked up what hadn’t splattered. The tree was quite tall, but we managed to shake down some of what didn’t drop. The grass cushioned the fall. The fruit was quite soft. When ripe, it had a purple haze on the skin and was sweet, eaten out of hand.