Are you in the UK? The measurement of the lining is only really an issue for women who are post meno- that is at least 12 months with no period. The cut-off point for those women is 5mm.
As this isn't you, of COURSE your lining will be thicker because you are clearly having periods, which are heavy or light due to being in peri menopause. During normal fertile cycles the lining can go up to 16mm.
I'm baffled as to why a woman at your stage of peri is having this procedure at all. It seems a complete waste of time. Biopsies are done for post menopausal bleeding or for more information is a woman has a thick lining and is post meno. They are not usually done on women having cycles and heavier bleeds during peri.
My advice and opinion is you need to ask for another opinion because the drs seem to misunderstand where you are in peri.
TBH I'm really shocked that a consultant suggested this. Do they think you are post menopause?
If you DO go ahead with the biopsy it ought to be straightforward. It's a very common procedure. I have had 2 both without a GA. I caught public transport home afterwards. It feels like a tiny nip for 1 second.
You may find that another dr is more adept at doing the biopsy without it causing you pain. You could have a local injection in the cervix, or wait until you have a period and that could make access easier.
It also sounds as if you may have vaginal atrophy and the weeing is part of that. Bladder pain or pressure can be caused by low estrogen levels. The treatment is vaginal estrogen cream that works really well.
Personally I'd question the whole need for this anyway. You don't have to submit to something.