Tricky - if she is quiet and getting on with work, then yes they're likely to assume she is ok even though you as her parent know her better and can see she is struggling. Schools can put things in place prior to diagnosis, since it isn't always easy to get a diagnosis done officially (for time and cost reasons, as you've probably already found.) Sensory overload will be a difficult one for them to change though, just by the nature of schools, classrooms, and other children. I have taught autistic children in a class of 30 at a state school, but each child's needs are so different that I think it depends very much on the individual. And from what you say she doesn't sound very happy at all 
Sorry if you have already considered this, but your other options are independent schools where the class sizes are smaller (although obviously very expensive), or potentially even a special school, although I hear they can be hard to get into. I have also heard from other MNers that the academic levels can be lower in some, although the support and social skills are more suited. However this could be addressed with extra tutoring if needed. The main thing is her happiness and confidence.
The only other thing I can think of aside from home schooling is to look for a very small state primary with no more than one form entry and preferably with small class sizes too. You'd obviously need to be sure that they were highly supportive - best ways to check are to read their Ofsted report and see what that says, and then book an appt with their Special Needs Co-ordinator specifically to talk about their SEND policy, how they would expect to support your DD and what they can offer. You could also ask them about autism in general - what do they see as the main area of support needed? (This will help you gauge how aware and how willing and also able they are to care for her.)
Also, how about ringing: www.autism.org.uk/ for some more help and advice? You could talk to other parents who have been through same, or see what the society recommend re your options.
And I'm so sorry that you and DD are having a very hard time right now, and hope you can find some support!