congratulations, twins are such a joy!

Of course how you want to birth remains important, and when you are higher risk it can be even more important to trust your caregiver. My suggestion would be to make an appointment with your local public hospital to discuss care options for you, and find out if there is any way you can enter a caseload midwifery team so you have continuity of carer and the knowledge that a known caregiver will form part of your decision making team. You may find there are midwives with a specific interest in multiple births and who will be delighted to caseload you. If your private midwife is still willing to support you as a doula then you may find this is all the support you need.

For my own twin birth I made an appointment with my designated OB in an NHS hospital and we came up with a plan that involved midwife led care with an OB to step in if certain conditions were met (preterm birth, bleeding, etc). If everything remained normal they he was happy to stay away because he knew I had made informed choices and that I would be "reasonable" if there were complications. I expected a fight, I did not get one. It's worth asking the questions and if you get a yes answer from an OB at the public hospital then ask them to sign your birth plan and detail their name so that subsequent doctors will see your plan and know it has been discussed and okayed by a senior member of their staff.

Good luck and I hope you find the answer for your care that you are looking for.