I think you need more estrogen. I have been where you are, when I first stopped desogestrel POP and went on HRT. What was interesting is that, at first, it seemed to get worse or more intense with each estrogen increase. I ended up in A&E after an entire night with zero sleep, just shaking through the night with heart beating fast. I was really wired.
I then thought I needed less estrogen, not more. That led to stopping the HRT, but the symptoms remained. Then I re-started it but was very very gradual about increasing - and this time, it seemed to do nothing at all. This was where I finally realised that I wasn't absorbing it very well and needed A LOT more.
When something sudden changes (for me, stopping desogestrel after 10 years on it, and starting all 3 hormones at once - for you, the radiotherapy) it throws everything into turmoil. It leads to huge fluctuations which in turn causes these symptoms. And no, it doesn't affect everyone the same. Otherwise there would be many more of us. But there are enough of us to know this is hormones...
Are you taking progesterone? How much and is it continuous? I would see if you can increase progesterone. It didn't help me, but it is the calming hormone and it has a sedative effect so it might help you.
Only increasing estrogen has helped me. But I know where you are, where it feels like it might be making things worse. The fact is that you are only on a 50mcg patch, which is the standard tiny dose they want to give women to have the bare minimum for symptom relief post-menopausally. But you have been thrown into sudden menopause and you highly likely need a high dose, because that is typical for women with sudden onset menopause. That's not even considering any issues absorbing it transdermally. Have you had estradiol blood tests, to see if you are able to absorb any of this 50 patch?
Estrogen functions like a neurotransmitter in the brain. When it cuts out, your adrenals try to compensate for it by making more adrenaline and cortisol. This leads to you not being able to sleep and lying awake with your heart pumping hard.
I would get some supplements which reduce cortisol and take them in the evening around 7-8pm. Ashwaganda, Alpha GPC and zinc are all good cortisol reducers and can be taken together.
I would really recommend you having a private consult online with the Newson Clinic so you can get access to the higher doses you likely need and increase very gradually. But be warned that it also takes time....