The school does not seem to aknowledge her joint problems and other issues
Daft question but have school been told? Did you send her EDS dx to SENCO in 2016 and ask for adjustments for her for PE and other activities? Did it list things she struggles with and the recuperation time necessary after any different movements? Do school understand how it impacts her daily living? IME schools "Don't see" so many things that affect our kids, even when they've had the medical reports.
Initially I'd ask for a meeting with SENCO, explain your DD's physical difficulties and see what options are available. I'd push for home-study whenever possible, if that's allowable.
My DD sounds very similar in presentation. Her fatigue levels and pain for days after doing things everyone else did without pain (e.g. putting on a seatbelt in a different car so using different muscle movements) hampered her so much more at school than her ASD and co-morbids.
no vitamin or mineral deficiences We were told that too as results of blood tests. One thing that helped my DD's fatigue and all-round "disposition" was/is still taking Aldi's A-Z Multivitamin and Mineral supplement, it's one tablet per day. After the first month there was no change, but after 6 weeks she was noticeably brighter and said she 'felt more like herself than she had done in a long time' Don't get me wrong, it hasn't stopped the fatigue or brain fog in their tracks but it has made a real difference to her. For once it's something that doesn't break the bank to try it!
She also takes a 'Better You' oral VitaminD 3000iu spray daily.
Other things to consider after you've sorted school for her exams - Have you applied for a Blue Badge for her? Have you applied for PIP for her? Would an EHCP help her in the future, they last until age 25.
For PIP, up to date medical evidence is good to have, but the 2016 diagnostic report should outline her difficulties, so that's something.
You can ask for her to be considered for a Blue Badge, maybe your local Council can do an assessment for her, they all have different rules so it's worth asking. Ours assessed my DD when her old badge had expired and her PIP was waiting for Tribunal decision. They renewed it and their report was also useful to submit as more evidence for the PIP mobility component.