FWIW you might NOT need to jump through as many hoops as you think.
I would advise you to try and get a referral before you even try and get pregnant to get this on record. This IS possible. You can get assessed as having tokophobia and get a CS agreed in principle before getting pregnant on the NHS.
Do some research into where has good mental health support in maternity. Not all hospitals have any support at all. There are specialists in birth fear out there. They are few and fair between but they do exist and they are becoming more common. Find somewhere that has experience of dealing with women like you.
How do you do this? Look for newspaper reports on the subject or which hospitals are doing research into the subject. Check hospital websites for information. Ask your GP. Ask friends. Ask MN. Awareness of this is becoming much more common than it was in 2011 when the current NICE guidelines were written. The bonus in doing this, is it puts you in control and offers you a bit of reassurance before you even start by arming you with the knowledge that it does not have to be a battle.
Its worth your while to at least attempt this, whilst also making plans to save - just in case - as a back up. It could save you a considerable amount of money that you can then invest in your child's future. If you are not pregnant then some of the stress of the process is already removed.
My other reservation about going private is it does not help you address your underlying fears and anxieties which are not just about your past experiences - they relate to your trust with HCPs in general and fear of the system. You need to think about how you deal with this too. And how this could affect your child. Your wider mental health is an issue here. This is something I never realised - nor appreciated until I went through it myself... I think its an essential part of your health care than money won't buy you.
As I say, I've been there. I have done this.
You sound like you are going through the same thought processes and decision making to me, that I went through 5 to 7 years ago. The idea of dealing with the NHS was one that I refused to even consider for a long time. Practically though, it really wasn't an option for me for go private as at the time there were no private options outside the South East at all and this was ultimately what forced me to rethink it and look for an alternative plan to the one I originally envisaged.
Please also be aware that having a private female obstetrician isn't necessarily going to solve your concerns either. They will not be the only person in the operating theatre and they may not be the only person who carries out procedures during the operation. You would need to have an all female support team too. There are also certain circumstances in which you might find things not going to plan - what happens if your obstetrician is dealing with another patient if you go into labour before your CS date? What happens if you have a medical emergency in which an intimate examination is unavoidable? Its rare but its a possibility you need to consider how you will deal with too. (I never had one). Also who is going to deal with you post-op? You will have a catheter that will need to be removed. This is extremely unlikely to be the responsibility of your obstetrician.
Even if you pay for it, you STILL can not guarantee that you will end up with the birth you want to be in control of. This is why building a relationship of trust with the culture and staff of the institution in general, rather than just a single individual, is an important part of the process, in case things don't go quite the way you want and because your care will be the responsibility of a team.
This is why you need more support than simply buying a CS. Its a bit more complex and the issues and concerns you need to think about are perhaps wider than you think now.
Ironically the best person I had was in fact a senior male midwife. He was the one who arranged my CS and listened to why I was so desperate to do this. One of my fears was the stress of the process itself and he recognised and supported me with that part of it.
I now have a 2 year old boy who is awesome.
The NHS were brilliant for me. I had a CS agreed in principle before I got pregnant. I had a CS date by 16 weeks. It is possible.
Whilst I appreciate care within the NHS is patchy and many places do have a culture of hoop jumping that is NOT the case everywhere. I never faced any such hoops as it was recognised early by people who had experience that I had a genuine and legitimate problem for wanting to go down this route. Stating I had a fear of hoop jumping also made it more difficult to do that as well as starting out with a referral to someone senior with enough authority to do that direct from a GP, rather than being referred upwards by a midwife once pregnant. It changes the balance of power and what agendas you might face, which I'm sure given your circumstances you will find helpful.
Getting properly diagnosed as having particular mental health needs for which you are entitled to the most appropriate care - which can include an CS from the offset - is your key.
Good luck. I hope my experience offers you some incite and some things to consider that you might not have and offers you a bit of reassurance that it is possible to have a very positive experience through the NHS in difficult circumstances. There are things you can do to increase your chances of getting this, if you are flexible about which hospital you want to give birth at rather than simply going for the one closest to you. Knowledge is power.
I have been on these threads a long time and read a lot of threads from very desperate women, who want an ELCS. Hoop jumping happens, but there are lots of places who are a lot more sympathetic than you might think and many, many women post here surprised that they are NOT subjected to it when they have been very fearful that they will be.