You absolutely shouldn't need to use private health care - however not all GPs are well versed in Menopause or perimenopause symptoms and treatment so it can be a bit of a battle to get listened to (some GPs also start by prescribing anti-depressants which is not the guidelines they should follow - but that's another story). So before going to the GP, do your research and come armed with facts and a good understanding of your symptoms (keeping track of sleep, hot fluhses, etc in a journal can be helpful).
Talking to other women going through the same thing has also be really useful for me - if only to feel like I wasn't going crazy or imagining things.
This all really helped in the process of confirming that indeed perimenopause was at play and more importantly get the all important prescription for HRT. This was over a year ago and has been life changing for me. I now feel like i am back to being myself again.
One thing to note: as hormone levels fluctuates constantly, test results can come back normal but it doesn't mean you are not peri-menopausal as the test only measures the levels at the time it was taken.
See below a few useful resources:
https://www.lattelounge.co.uk/
https://www.balance-menopause.com/
https://www.newsonhealth.co.uk/resources/#factsheets
https://mysecondspring.ie/
All the best on your journey - you do not have to suffer from this - if indeed hormone deficiency is what is causing your issues there is simple treatment(s) available.