If you had a child that was utterly miserable in your home, this had been going on for years, they were desperate to live with their other parent and constantly requested to go, and were unwavering in this, would you let them go? One of my stepchildren is so desperate to leave home they are going to walk on their 16th birthday. Right in the midst of their gcses. Surely mum would be best placed to let them leave now, the year before? In an ideal world the child would be put first but I'm not sure that's going to happen. Child is going to pack a bag and walk literally that day.
(I do have experience of this as one of mine went to live with Dad for a while as a teen. It devastated me but I knew it wasn't all about me)