It's so hard when you know your DC aren't happy isn't it? I used to think the baby/toddler years were the hardest!
My DS has autism too. He's in Yr11 now and is finishing his GCSEs. A few things jumped out to me in your posts.
Firstly, I would say - if you decide to go for an EHCP at some point, don't get fobbed off by the school/LA claiming that she's too academically able to get one. My DS is predicted mainly 8's/9's in his GCSEs. I know DC at Grammar schools with EHCPs. They are designed to support for all kinds of barriers to learning but also for social, emotional difficulties too.
I was told by the school SENCO that my DS would never get an EHCP as he's too able so I applied myself and he was granted one.
You are correct in that it could take a long time though - if you have to appeal refusal to assess and to issue you could be looking at two years. That would be in time for KS4 though.
You say that 'the support emotionally and socially that we were promised hasn’t really materialised' - sadly, again, without an EHCP that is likely to be the case at most schools. Schools are overwhelmed with students needing support and being brutal, if your DD is a 'quiet, good' student, there's a good chance she's going to get overlooked.
The other thing about applying for an EHCP means that she will have a needs assessment and will be seen by an EP, a SALT and an OT who can help to identify her difficulties and make recommendations. (Not just academically but also it covers things like making friends, helping to hold conversations etc.)
Moving schools and hoping things will organically get better is unlikely. However from your perspective, how realistic is it to be driving 2 hours a day for the next four years? (I've been having to take DS in for 8am during his GCSES so that's something to bear in mind.)
Have you met with the Senco at DD's school? Even though your DD doesn't have an EHCP, they do have a duty to support SEN. Are there lunchtime clubs she can go to? Even if she doesn't make actual 'friends' there at least it's something to do.
Could she become a library monitor?
I would also say in terms of Facebook posts, like all reviews it's most likely to be a handful of people who have the strongest opinions. I would try and make an appointment with that school's Senco too and see if you can look around the school for yourself.