Change your GP. No one is 'too young' to begin menopause. Occasionally girls h ave a few periods then nowt; rarely girls never menstruate. How early is one supposed to be for menopause to be too early?
NICE guidelines suggest HRT rather than ADs etc.. However, if mental health is a dangerous issue, ADs should be tried for 8-12 months with consideration of appropriate HRT - going on symptoms rather than blood tests.
I suggest that you ask for a referral to a dermatologist with knowledge of hair loss. Alopecia can happen at any age and for no particular reason. Is your hair dropping out in clumps, mine seems to be coming out more when I wash it. No gaps fortunately. As well as a referral to a dedicated menopause clinic as there are waiting lists. If by the time an appt arrives you can then decide whether it's appropriate.
Don't give up. Is there a Nurse Practitioner in your Surgery who may be knowledgable about peri menopause? These years leading up to the last period can be an upheaval as hormones rise and fall! Let us know how you get on.