Katy, from a laymans perspective, you're absolutely doing the right thing.
The trouble with our lovely NHS is limited resources. I have found that healthcare professionals frequently try to cut costs by avoiding scans/blood tests/other costly procedures as much as they can. When we lived in America, the first thing the doc would do were tests, to diagnose. whereas here doc diagnoses through symptoms/some form of trial and error
I would categorically not accept the diagnosis until proper tests are done, don't let them fob you off until you're satisfied that you know what's wrong with her, or at a minimum what's not wrong with her.
We have private insurance. Afaik, no private gp as such. You need to have an idea of who you want to see (perhaps call for paed, neurologist, opthamologist recommendations on MN, I gave no idea who you need to see), and then go to your GP and request a referal. Make it clear to your GP that you'll pay for it privately. Our GP is reluctant to refer to specialists, but as soon as we mention it will be done privately, we get immediate referal.
hth. Good luck and keep us posted on how you get on. It's a disgrace the way you're being fobbed off without any proper diagnosis done. an utter disgrace