Sorry to hear your child is burntout. Both of mine hit burnout in Dec 2024 and crashed out of MS; the older one (your child's age) still hasn't returned to school but is now in a much better place; the younger one (13) is still struggling, unfortunately, even in a lovely SS, and has spent most of the summer hols inside.
It is really hard to see your child in burnout and our instinct as parents is to try to get them support and fix things. However, I found the absolute best thing we could do for our kids when they were at the bottom was to remove all demands -- no pushing them to go to school, or to see doctors or therapists, or to do activities or socialise or help around the house. It is incredibly hard (and I self-doubted/backtracked and messed up many times), but ultimately I think it was the only thing to really help with recovery in our case. It took a couple of months before they started to engaging with self-directed activities (reading, doing art, etc) and about six months before they could see anyone outside our immediate family.
Naomi Fisher's Guide to Teen Burnout was really helpful at the time.
We were fortunate in that our SENCo was very supportive and authorised all absences, but I think that's rare. If your DD is not going to be able to return to school in September, then I'd request section 19 AP as soon as possible (you might have to wait until the start of term to do that). It could be that she finds it easier to engage with learning or other activities once she feels that the pressure to go back to school is off.
I would make sure to document every conversation with school, GPs, etc in writing so that you have a clear paper trail.
And be gentle and patient with yourself as well as with your child. It's really really hard.
Do you think DD would be able to do an online AP or school when she feels ready? Minerva, Kings Inter-high, Academy 21 and Gaia Learning might be worth looking into -- but again, if you can manage to just give her space and time, you might find it works wonders. Both DH and I work from home regularly, so we were able to arrange our schedules so one of us could always be home, but I know not everyone has that flexibilty.
FWIW, my 15yr old now says that leaving school was the best thing for him. He understands more about himself, his interests and is planning to start his GCSEs early this autumn through EOTAS (or home ed if the LA won't fund it -- we're still waiting for his amended EHCP). He also recently started volunteering with a local arts organisation, which he would never have been able to do if he'd been putting all of his energy into surviving the school day.