Hally, you have my deepest sympathy as this is also my main issue. I have been through every variation of insomnia over the last few years, from not being able to get to sleep for hours, to sleeping but then waking early, to crappy light sleep where you are just trying to keep yourself under....
And like you my serum estradiol was 330pmol when last measured and I was very symptomatic.
Luckily I'm with Newson Health and they recognised that some women need higher E levels especially during peri. Two different doctors there advised me to try to get between 450-650pmol and to increase above licensed dose if necessary.
I was on the gel when we first tried this. I got up to 6 pumps, which got me to that 330pmol. Then I continued to increase up to 12 pumps (yes, 12 per day). But serum levels didn't increase, they were still around 330pmol. So I was unable to absorb more gel.
So then I reduced back down to 6 pumps and introduced patches and increased those. I'm now on the 6 pumps of gel plus 2x 200mcg patches. I don't yet know what my serum levels are as I just reached this goal dose about 4 weeks ago, and I'm trying to test during my period to see what I'm getting from HRT (not what my own body is weirdly contributing when it feels like it).
My sleep has *significantly* improved. I went from the 2-3 hours a night like you, plus shaking all night with inner tremors and my heart beating hard and fast in my chest, hypnic jerks every time I began to fall asleep.... to getting about 5-6 hours a night. You can't function at all at 2-3 hours. You can function at 5-6, although it's not optimal. I go to bed around midnight and I still tend to wake in the early hours around 4-5am usually - and then I find it hard to go back to sleep, but after a couple of hours awake, I can often fall back asleep for another couple of hours. I am also getting some deep sleep again - both according to my Apple Watch and to how I feel. I was unable to get into deep sleep when things were at their worst and was just dozing.
I may need even more E, but I will test first. I am also on high dose P: No P for days 1-5, then 300mg days 5-14, then 400mg the rest of the cycle. Partly because it helps with the sleep and to offset any worries about high E doses.
I would highly recommend you go to a private menopause clinic which is happy to prescribe higher doses. You are unlikely to get prescribed above the licensed dose on the NHS.