Hi, (have read your other thread as well as this one) my suggestion would be rather than allowing her education to be something that overnight changes her relationship with her dad, and starts WW2 in a ?who is right/responsible? and potential tug of love way given that he seems to have 50% responsibility, you enrol her in an internet school, for the moment.
It will give everyone a breathing space, stop you feeling desperate (not good for anyone) about how to go about things, allow everyone to come to terms with not doing what is seen as ?normal?, while continuing a structured curriculum with other children for the moment, in a safer environment.
Her dad probably won?t like it either, but he?ll be hard pressed to do much about it other than refuse to let her log in at his, in which case he?s disrupting her education. If he gets arsey he?ll find it much harder to make a deal out of it, and it seems to me that your daughter having to see less of him to sort out educational problems, and stop him and mum going at each other over it, isn't a good thing for her and is storing up many problems.
It?s quite normal for children to enrol for short times and doesn't have the same problems as going to a standard school short term and moving on, and in cases where parents end up in court is easy to get court recognition as ?suitable education? and prevent residency arguments because of educational differences claims. (I?m not saying that?s what would happen with him, just that you seem to feel he?ll go mad, and the response is she?ll see him less etc)
It would give you time to take stock, learn more about different ways of home ed, and give dad a chance to accept that whatever is happening, apparently his daughters education will in future be ?different? and he needs to learn more about the subject.
You may find she flourishes (Interhigh is particularly good with bullied kids) and wants to stay doing that, or you may find you want to do things differently and go for that, but all ways round, you reduce the ?out of the blue? high drama and impact on all relationships, potential for him to start legal action etc and can show you've thought through your DD's best interests.