Firstly this link
https://www.drlouisenewson.co.uk/knowledge/burning-mouth-syndrome-and-menopauseI am sorry to have only picked this up but yes I have and do experience this and it was diagnosed back in 2016 after referral from dentist to maxillofacial at hospital. Told I had Burning Mouth Syndrome known as BMS. Little is known of this, it flares in women and my mouth was on fire and lips so sore. Initially I was given prednisolone 5mg soluble tablets which had to be dissolved in small amounts of water. It settled down and in return I was told I had Dry Mouth.
Here goes with what I was told: the mucous membrane is an inner lining found in the mouth nose and down below also the inner eye. Me well I have the best of three dry mouth, dry eyes and VA too boot. (all diagnosed) and under control. Seems can be more in women of certain age!!!!
I do think the dentist referral is correct. Get prescription from Dr called Oralieve gel which helps with moisture in the mouth. Can be bought on Amazon as can their toothpaste. Also Salivix pastilles are on prescription or Amazon for mouth while never used there is something called Xylimelts that stay onside mouth if dryness bothers you at night .For the lips sorry but this was a nightmare for me.
While covid with mask it was horrendous. Tried allsort and a dental nurse suggested olive oil mixed with tiny amount makuka honey . Just a dab here and there on lips and it really helped. Plus had in house so no brainer to try If you wanted to go for a lip treatment have used La Roche Posay Cilcplast??
Do hope this helps and once on right track it will improve.