Sadly this is a really common situation for internalising autistic people, often girls. They do OK at primary, are conscientious and no trouble, express school based anxiety at home but not in school, and then don't manage the transition and wheels come off somewhere around year 7 / year 8.
Is she still attending? Emotionally based school avoidance (EBSA) is common in this cohort.
One issue is that indie schools do not have to follow the SEN code of practice by law (though many do) so as a parent it can be difficult to find suitable leverage. The school may have profs it has links with, though it's likely you would need to pay for prof reports etc.
With a large health warning as obviously I haven't met your DD and don't know the full situation, but I would imagine she needs an ehcp first, and then a discussion with the EP about what type of setting would best meet her needs. The statutory process would then include a consideration of the range and suitability of provision available for your DD.
I guess the key points to make in a request (model letters can be obtained from IPSEA website) is that her mental well-being is suffering, she is autistic , and at risk of EBSA or becoming NEET (not in employment, education or training).
The key question for you would be, is this a situation that more or different support would remedy at this school, or does she need a school with greater expertise or more flexibility around curriculum?