Keep a diary of your DD's behaviour, Nigella. Daphne Keen diagnoses through a clinical interview, which means, as well as observing your child herself, she asks pertinent questions to see if your child fits the latest criteria. It's crucial to have examples of relevant behaviour.
My son is also 14 and has had no end of appointments, poor boy. I didn't want to mention Aspergers to him until I was certain, as he has already had to cope with other diagnoses, so I simply said Daphne Keen might be able to find out why he gets so anxious about things. That was enough for him. Could you do a similar thing with your DD? Is there anything about her behaviour that she would like to change? Maybe you could say that this doctor might be able to help her cope better in some way?
Also, the diagnosis will be of High Functioning Autism, if your DD gets one, as the term Aspergers is no longer used in the latest diagnostic criteria (DSM-5), but when Dr Keen said DS fitted the criteria for HFA, I immediately said, "You mean Aspergers?" and she nodded. In her report, the diagnosis is for HFA. I called it Aspergers to DS1 at first, but he must have googled it and he later said to me, "I'm autistic, aren't I?", which took me aback. He seems to have accepted it quite easily.
I really 'get' that feeling you're describing of there being 'something' different in your child. Sometimes DH and I felt like we were going crazy because we could see this 'something' and other people didn't. The point is, you need a trained eye (like Daphne Keen) to see it.
Incidentally, like your DD, my son also does not seem to have obvious sensory issues (although his refusal to wash his hair, cut his nails or visit the hairdressers could be due to this - hard to tell with a teenage boy). The lack of sensory issues doesn't mean your DD DOESN'T have Aspergers. It may be that her difficulties lie in other areas. My son is good at reading expressions, for example, but his severe difficulties in other areas still warranted him a diagnosis.
Don't worry that you are doing the wrong thing and 'meddling'. If it turns out that DD doesn't have Aspergers, at least you can cross that off your list. You may find that the 'something' you suspect is something else, but you are right to trust your instinct that there is a 'something', if you see what I mean. Parents are usually right, in my experience and CAMHS are usually wrong!.