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!)
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.
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.
When will she know if she has been accepted at the college? Will she need academic qualifications to get in there? I know a girl who is now 16 and has a number of GCSEs and A levels already and these were all done on her own whilst she was in West End productions but she is incredibly driven and talented. She knows she wants to go to UNi so needs them to get in. If your DD knows she needs her GCSEs she might manage all of the after school activities and the study. Will she be dropping some of the activities when she is at college- I guess she won't be studying them all there
Yes easy, but you'd have to start IGCSE courses anew. I'm on DD2 now with IGCSEs after having put DD1 through 10 over 3 years. We've always started them in the September and taken them in May/June, and this year I've done that with a 13 year old, so a bright, motivated 15 year old would be fine. You have some background research to do and an exam centre to find pretty fast, though! But you have time.
If she is motivated and bright, I don't see any reason why she shouldn't be able to self-study for 5 IGCSEs in a year. The time can be used far more productively when there is no travelling or "admin" involved. With IGCSE science the practical assessment is replaced by an additional short exam paper, so the practical part of it isn't necessarily a problem.
It depends what you and she are after really - can I assume you are focussed on the fact that she would be better off having 5 decent GCSEs before leaving school, as a fallback to her dancing ambitions?
We were in a similar situation with DS and music - in his case he could spend 2 years at an online school to get a "standard" set of qualifications, which I felt was important no matter what else he wanted to spend time on. It's not strictly necessary for HE, but I do think it is convenient to have them.
Would she be able to study while travelling? And/or while in London - am I right in thinking she is staying with family, so she'd hopefully have a decent environment to be able to get down to studying if she wanted to.
Would the school let her leave Friday lunchtime, as it's a vocational activity. Would they mark her assessments and let her sit the exams (for a small fee). If not, you might be looking at igcse. It's definitely possible in a year, but you will have to work out how. Maybe she could do a couple in jan and the rest in the summer? There are correspondence courses she could do.