It was the doctor that suggested it, when my other symptoms were sorted, but the brain fog persisted. I wasn't even having sex at that time. I see someone else now and they too are fine with prescribing it for it's other benefits. I think generally, most private specialists are open to prescribing testosterone, even if there isn't hard evidence that it helps with things beyond libido. Clinical experience tells them that it does, and it makes sense when you know what roles testosterone plays in women's bodies. Best to check first though, before you pay your money. Of course, there's nothing to stop you telling them it's for libido but I would prefer to be honest about my situation, so they can deal with it properly.
I think in the NHS, it's partly a financial thing, with the extra costs of having blood tests to make sure you have sufficient oestradiol levels first, and then monitoring to make sure your testosterone levels stay within certain parameters. For me, now I'm established on it, that means a once yearly blood test. If you go privately, none of this will be cheap of course, and it may take a while to know if there are any benefits. Haver you had a look at some of the testosterone threads on this forum?
For you, I think it might be worthwhile having your oestradiol levels checked anyway, which you can do yourself through companies like Medichecks. Were you any better with oral oestrogen? They can also check testosterone levels, including SHBG and Free Androgen Index, which you will likely need beforehand anyway (although many specialists may not accept the results and will probably want you to do tests through them). If you do it yourself, make sure you have a blood draw and not a fingerprick test, which are unreliable, and have the test done prior to your Oestrogel application that day.
Was your EDS problematic before becoming menopausal? Hormonal changes clearly have a big impact but also just the normal decrease in muscle mass with ageing, which you will feel more acutely with your EDS, although it sounds like you are doing what you can to mitigate that.