It does sound consistent with Aspergers (though it all just gets called autism now) to me so although obviously it could also be something else it certainly seems worth investigating.
My DD was Dxed at 4 but I didn't see it at all. Her only obvious symptoms were physical (she is hypermobile and was a very late walker - which is common with autism). Her nanny picked it up because she was late to play with rather than alongside other children, and although she had very advanced language in terms of vocabulary she didn't ask for things until very late. She also reversed pronouns maybe a bit later than most kids (so when she first started asking for things she'd say "do you want a drink?" rather than "I want a drink"). Once we started getting into the assessment process they also put quite a lot of weight on her not pointing at things until quite late. DD actually does quite a lot of imaginative play but again I'm told it's somewhat rigid (again, I didn't notice - but I can see now that I am also broadly a "lining up, anxious about change" kind of person so none of this stands out to me).
DD doesn't have obvious stims. She used to tense her legs when she was excited when she was little. Now she jumps if she's really excited but that doesn't really stand out as a lot of 4 year olds still jump when excited. But no hand flapping or finger twirling or rocking or head banging or any of the obvious stims like that. Autism can be really subtle but it sounds like your CAMHS is not very up to speed.
By contrast DS is well behind on language (at nearly 2 only has about 20 words) but is a great communicator (will point to things he wants, will point in his mouth if he is hungry, will shrug his shoulders and put his hands up if he wants to know where something is, etc.) DD never did any of that (but she was my first so I didn't notice its absence).
It is very common for "high functioning" kids (maybe girls in particular) to hold it together very well at school but it's a lot of effort and can mean they are then distressed at home where they can safely express themselves.
A lot of the ways you would parent (and that teachers should teach) should be the same regardless of whether your DD is autistic or has anxiety. Knowing what is coming up, having choices (within acceptable alternatives) so she has a sense of control, will all be helpful either way. Thinking about any sensory issues could also be helpful. Have a look at the Stuart Shanker website (Self Regulation) for some ideas. There is also a free training module on autism on girls on the NAS website that may be helpful (it's just free for the first year - I must get around to doing it myself!) It may also be worth printing off some materials from the NAS website on autism in girls that you could take to the CAMHS appointment with you, along with notes on your observations?
You might also find Tania Marshall's list of signs of Aspergers in pre-school girls resonates - though I suspect if your CAMHS person is unenlightened a "random" article from the internet is not going to help (although you can see she is actually well recognised and respected in Australia), and it may be better to focus on materials from the NHS/NICE/NAS for the CAMHS appointment. taniaannmarshall.wordpress.com/2013/06/22/first-signs-of-asperger-syndrome-in-young-girls-pre-school/
Is a private Dx an option? In terms of NHS, is CAMHS the only route to a diagnosis in your area, or could there be options to get a paediatric referral through your GP?
YouTube and Twitter are also great resources for learning more - many adult autistic women there who can give insight and advice.
You could also try a book called "All my stripes" - about being different (specifically mentions that the character has autism though my DD has read it and doesn't seem to have picked up on the reference), and maybe "Something Else" and "The Wonder" by Faye Hanson (which are more generally about being different). I also like the Sally Potter books on emotions and friendships.
Ideas for ramping down the anxiety: don't try to make her eat things she doesn't want to. Provide different things and praise her if she licks or tastes but no pressure. Don't criticise her play. If she wants to line things up or set scenes that's fine. Try to get involved and flex things just a tiny bit. E.g. will she let you put one character in her scene, or take on board an idea you have for how to play. Use e.g. turn taking board games to practice turn taking. Get books about relationships and friendships (DD has just started to enjoy the "Magic Kitten" series and they have quite a few friendship problems in there that I think could be good to talk about as she gets older). Do a diary (maybe just once or twice a week to start with) where you can talk about the best and worst things that happened that day/week. DD can draw pictures and talk about how things made her feel. You can talk about different things to try when tricky situations come up in the future.
The good news is that if your DD is parented and taught in the right way I don't see any reason why she can't be very happy and successful, and the fact that you are already so tuned into her needs is a brilliant start. Have a look at some people like Invisible i, Sew Many Books, Clodplaye, Connor Ward, The Aspie World on YouTube. There are many more, and yes lots of them have faced a lot of challenges growing up - but we can do so much better for the next generation. These are all brilliant young autistic people who can show you that the future need not be negative with the right understanding and support. Also worth looking up Purple Ella (and her friend Roz in the "autism in company" videos), Rachel Lucas (the author), Laura James (another author) and Sarah Hendrickx (all older autistic women).
Sorry for the essay but your post really resonated with me - I hope it's not too much!