How do I go about going private
If you want to consider a private consultation Emm225, have a google for private hospitals in your area & they should list their consultants, together with their special interests. I'm not sure whether you're thinking Gynae or Gastroenterologist since you mentioned both bowel & gynae issues. Either way, if they don't state fees or you need additional info as to who would be right for your concerns, waiting times etc (likely to be minimal) you can ring the hospital's admin/patient services section or the consultant's secretary to find out what you need to know, before booking. Many accept self referrals these days, some may still ask for a GP referral. If you are wanting a dedicated private menopause clinic rather than a general gynae or gastroenterologist, again you can put that into your search engine for the area you live in.
Alternatively, if you can face it, you could try your GP surgery again, stressing that you're worried. At very least they should be able to tell you the outcome of the stool test if it was arranged through them & how to progress it if it was through the postal screening service. Many of us do have probs with unscheduled spotting/bleeding & bloating on "no bleed" conti HRT regimens & it may well just be a question of tweaking yours, but if your instincts are that this needs further medical input I would go with those.
Wx
Later edit: I've assumed you're on continuous HRT, but rereading your original post you mention "certain times of the month" so are you in fact on sequential HRT but bleeding at odd times rather than just getting a withdrawal bleed after your progestogen phase? Sorry if I've missed something, but the uncertainty over whether you may not be postmenopause also makes it harder to know what's going on, as you say.