I'm not sure if you need to be perimenopausal to be prescribed that. The fact that your blood tests are showing that they're low is not good. I'd insist to be referred to a gynaecologist if I were you. They're not all confident with dealing with hormones but some are.
There is also no hard starting line for perimenopause, and it affects women differently. Its not like you reach 50 and then it accelerates massively in a year - it could start slowly over a decade before that. Might be aches & pains, poor sleep things like that. It's not necessarily crippling anxiety or night sweats.
That being said, testosterone doesn't always fix libido, so be prepared for that. It might improve the actual physical sensation itself, but libido could be a mixture of different elements, estrogen and dopamine being two important ones.
Honestly I think that so many GPs are just full of nonsense when it comes to stuff like this. They can test you if you are within a (very wide) range of 'normal' for the average population, but they can't test for what's normal for you. For all your GP knows, 5 years ago your typical hormone profile could have been quite different. Now, it might still be 'normal', but it's just not working for you. None of them will acknowledge this because it requires more nuance and thought, and lot of them are used to thinking in boxes. They're also understandably very hesitant to prescribe things off-label(which in your case this would be).
If you don't have any issues with hypothyroidism and you're not on anti-depressants, and want to see if it is actually hormones that's playing a part in this, then you could try fenugreek. Health4all used to produce a good quality one but I can't find it on their website at the moment (it's worth asking them if it'll be back in stock). You could start with 500mg and see how you feel after 6 weeks (takes a while to work). Read the research on it - I've seen a study or two showing it raises hormones levels quite a bit but it might not be lasting. When it comes to the dopamine effect (which it also has), this tends to level out eventually. It's a good test though in my option. However, don't expect any regular health care professional to know anything at all about it - this is experimental.