If CAMHS recommended educational psychologist as a first step, could you try that? There are often local SN groups that can recommend - again, though, we did find that there are a lot of very passionate parents there who recommend specialists who agree with them, similarly we were very sceptical of some of the online reviews. Are you in touch at all with old SENCO? Also if you know of the good local children's hospital, you could Google the names of specialists there and see if they have local private practice.
One challenge we have found is that while a diagnosis helps to access more suitable support, it does also have a weight and it's one that's lifelong. That can be totally worth it if the issues are major, as they were for my DC, who was becoming violent and refusing school, but even in our case, where it's overall absolutely worth it, it's become apparent in surprising ways that putting a name on things can hinder as well as help. Googling symptoms at 9 concerns me a little, as while she may well know and recognise herself, it's also very easy to see ourselves on lists of symptoms and emphasise what fits and forget what doesn't. This is true even for adults (like with horoscopes) and even more so for children, it also means she's potentially going to be giving the 'right' / wrong answers when she's talking to a professional. It's good if she can see someone soon who can help her and you get to the bottom of this.
If she's doing well at school and you're not immediately seeking medication or major adjustments, one positive of seeing someone privately, and starting with a regular psychologist appointment vs an official assessment or a psychiatrist, is that it doesn't have to be passed onto a GP or have an official diagnosis right away.
You can then start with seeing whether her issues can be addressed for now with some CBT or coaching and then if that doesn't work or if the psych feels that there is likely an underlying condition Vs just 'traits' and that diagnosis/medication would help, you can go ahead with getting an official assessment.