As a parent of a child with very bad anxiety, social, and general, I really do empathise. DD has had counselling for four years now, privately, and it has helped, but she will always have anxiety, the counselling just helps you deal with it.
For appeals grounds, I don't think your DD is unusual, lots and lots of DC's seem to have heightened anxiety these days. So many are under counsellors it is like an epidemic. |So not sure whether this will be taken into account, but worth a try in any event.
Although she thinks she needs her friends, she really needs to be helped to cope without them, the counselling hopefully with do this. Remember, most friendships change and move in secondary school. Even if she gets in, she may not be in the same class, or the same sets. She may find they move on, find other friends, and this is also a big deal, a huge deal in fact, if you are relying on a group of friends, if that happens she may be really thrown off course.
This sounds all very negative I know. But, honestly, just in case the worst happens, try to get her all the help you can, pay if you are able, and get her to be more resilient. It's really hard, and very upsetting for parents, you will never stop worrying about them, but they do learn to cope eventually, and hopefully move on healthier and happier.