Hair loss is so frustrating and upsetting. I had some hair thinning in perimenopause, but turns out my ferritin levels were very very low, to the point that I ran out completely so might be worth having that checked. I changed shampoos, upped my iron intake, things stabilised, hair stopped shedding and started growing.
As for HRT, I thought that the progesterone was the culprit the first time around. My GP prescribed Kliovance last year. After my hair had been ok for quite some time, a few weeks after I started that round of hrt, my hair started falling again, like you I'd be in tears in a bath full of hairs floating around. I read that Kliovance has norethisterone which is testosterone derived and could cause problems such as hair loss. I asked my GP if I could try Femeston. That did nothing and my hair kept on shedding. So I stopped. I went almost a year with my miserable menopausal symptoms, constant anxiety, dark moods, extra tiredness (I am always tired anyway as I have fibromyalgia). But my hair stopped shedding.
Last summer, I couldn't cope anymore, having read a little more about HRT and people's experiences, I asked the doctor if I could try Oestrogel and Utrogestan. At first the GP I spoke to gave me progynova and utro. Then my own GP switched me to the gel. I could not tolerate the daily utro, it floored me with fatigue and other side effects. My hair started shedding too, again. So I decided to try utro on a cycle. I've been on Oestrogel only for a couple of weeks now, a lot less tired without the utro daily, but my hair has started shedding even more than before.
All this to say that your problem might not be the progesterone, but the estrogen It can help with hair loss, but from what I've read it can also cause it. It is a tricky one

I went to a dermatologist a few years back when I was concerned about hair loss and he didn't even examine my hair or scalp, didn't do a hair pull test, didn't loo at my blood tests, he just said I was getting older and it was normal.

How helpful.