Much as I'm a huge fan of home ed, I wouldn't be taking a child out of school at this age who wasn't absolutely desperate. From my experience of the DDs and their friends, doing them out of school is much harder work in a way. IGCSEs are very exam heavy, there's no controlled assessment or coursework and some of the exams are well over 2 hours long with at least 3 of them per subject. Yes, there is less time wasted than at school and maybe more relaxation, but you lose all that teacher support, too. DD2 is doing just two this summer and we have done some sort of work or studying every day for the last 6 weeks at least, no days off. You have to learn everything as you have no idea what will come up whereas DD1 in college doing A levels this year seemed to have huge content hints and her 4 AS levels were definitely less stress than DD2's two IGCSEs have been. Home ed is wonderful, but you need a very comitted child to do the exam route -they have to 100% want to do it for themselves. DD2 has got through the last 6 weeks because she would never in a million years agree to return to school.
I think she needs at least one evening of no dancing . What type of dance does she really want to pursue ? Focus on what she needs to audition , are there any studios/halls nearby you could hire for her to rehearse by herself or have a private lesson at, to cut her weekly, travel time and maybe do an intensive course over the summer holiday to prep her.
It sounds like you've managed at least a bit of compromise - it's very tricky with teens who are so motivated/obsessed.
A couple of thoughts - is there a dancer / dance teacher who could talk to her about the importance of rest etc for a young dancer? She might take advice from someone in the profession that she'd ignore from 'just' her mother.
And you could try discussing flexi-schooling with the school - we persuaded DS's school to let him have one "column" of his timetable out of school for a year to work towards Grade 8 music. The important things for them were that he'd be working towards something that was a recognised qualification, there'd be supervision at home, and so they wouldn't face a flood of similar requests from others (I think the last was the clincher from their point of view!)