I think it would for sure be worth you exploring a higher estrogen dose via a private clinic in that case.
As Dr Newson has been saying a lot recently (and as recent research shows) basing things on a certain dose is insane because there is an 11 fold difference between women in terms of absorption of gel and I think it was a 7 fold difference in patches. So why is everyone obsessed with the dosage applied via patches and gel, when absorption differs so much from person to person... Your 4 pumps might be the equivalent of someone else's 2 pumps!
I see Dr Olivia Jones at Newson and would recommend her. I previously saw someone else there who I wouldn't recommend. Just as with any clinic, there are going to be great doctors and not-so-great doctors. I don't think any Newson dr would refuse you more than 4 pumps of estrogen tho
When you see them, it's totally up to you whether you get the meds from their pharmacy as well or via your own GP. Sometimes people get Newson to write to their dr and explain what has been prescribed and some NHS GPs are then happy to continue prescribing that if it is working for you and you're settled on it. Othertimes everyone throws their toys out the pram and the NHS GP still doesn't want to do what some private GP has told them to do, and still refuses because they can't admit they are wrong or know less. In which case you could still get SOME meds from your NHS GP (they will think you are on the lower dose... ahem) and top up with additional from Newson pharmacy. You can order as much or as little as you want from Newson.
When I first saw them my plan was to get as much as possible on the health service and just top up from Newson. But frankly the health service has never had all of my prescription in stock locally. At first it was the estrodot patches, then it was testosterone, now it's utrogestan... will it never end... Whereas Newson just always has everything in stock via their CloudRX partner pharmacy. Utrogestan is limited to 2 months supply at a time at the moment but they are getting it out, mine just arrived a bit later than the rest of my meds. You get whatever you order in the post about 1 day after you pay for it. If you need more, you can just go online and order it from their website as long as you are a current patient. To be a current patient, you need to have a consult once a year - it might be slightly more frequently at first if you are going through a problem and need to stabilise meds etc. By the way, the most expensive med seems to be utrogestan(!) so if you can get that on the NHS you'd save a lot even if you got the rest privately!