I think the only thing to be done in these sorts of situations is put the child in the centre and do the calculations based on their gains, and their potential losses.
I removed my son from a state school years ago. I am not anti HE. But as with any educational choice, the child's needs have to be the absolute priority.
People are not backwards at coming forwards and making bleak warnings such as "SS could take you away!", "it's illegal !" even to children. They don't always limit it to when you are around to deal with them on her behalf either. Where there is agreement between parents that can be handled most of the time to avoid the child worrying themselves sick.
But where there is strong, ongoing disagreement between her parents over her education... it may be hard or impossible to convince her that she is not at risk of dramatic changes, or bad things happening to her/her mum, especially if her dad does escalate things.
It isn't always a cake walk, both for parents and kids, to shift to HE. There is a whole other level of stress and angst included in your situation. Which could be for an extended period.
Given all that, are you absolutly sure that what your daughter will get out of HE is going to be worth all and any emotional costs she might have to pay if things turn quite nasty and/or move to a full blown legal dispute between you and her dad ?
If you are not convinced the answer is "yes, totally worth it, still a signficant win overall for kiddo", perhaps it might be worth shelving HE for now. Find ways to supplement her school education. You can always revisit the question in the future to take its temperature and see how the land lies with dad then.