If it was me I wouldn't ask for a ring. But if he want's to give me one as a present I'll take it. Now if later I found out it belonged to somebody else, I really wouldn't mind.
Your ex sounds agressive, she reminds me of my sister, who used to put on a show everytime the father of her first daughter came to pick the kid up for the weekend.
Perhaps a third neutral party could drive your daughter back and forth, she shouldn't have to witness that awful behaviour.
RE: Polyamory
Feel free to start a thread on that subject.