Docs know very little about perimenopause....I had worse insomnia peri than post. Was offered anti-anxiety meds but didn't want them, so just pushed through which was incredibly hard. Brisk 2 mile walk per day helped sometimes, others it didn't and I just used the time awake to work!
The essential oil Clary Sage is a natural sedative (so much so there is a 'no alcohol' warning) and if you put it in a bath and some rub in oil it will zonk you out. Again, sometimes that didn't work so you need to try a few things. Progesterone also put me to sleep, but I felt drugged the next day.
Box breathing can help - in 4, hold 4 out 4, hold 4.
It's a really tough phase but it does pass - I am 3 yrs post and sleep is fine now (just my insides are falling out!