AliBaliBee....
This is a bit of a confusing situation you're in there.
You are definitely not too young for this to be menopause/peri-menopause. I've been on HRT since 42 (now I'm 46). And I've had symptoms since age 38. If your GP is telling you that you are too young, you need to find another GP because that's just ignorance - and there's no excuse for ignorance in a doctor when it comes to something which over 50% of the population are going to be dealing with.
Secondly, when it comes to insomnia (which I have had a heck of a time dealing with and has been one of the worst things for me), for most women it is caused by low/fluctuating estrogen. Not progesterone. That means - you need systemic estrogen. If your doctor has just given you local estrogen cream, that's lovely for your vagina - but it's not going to be doing anything systemically for you.
Thirdly, the vast majority of birth control pills use synthetic hormones. These cannot replace the hormones your body used to produce because they are completely different, chemically and structurally. There is one pill called Zoely which uses body identical estrogen (the same as in HRT) and a synthetic progestin and another called Qlaira (which is used less often) but besides those two, you are looking at synthetic hormones in BC. If you really want to stick with birth control, try Zoely.
Otherwise, move to HRT. Really it sounds like you need to change doctors....