Hi Stef, I was interested in your post as it is about paper based assessments. This is because we managed to get one for our daughter last year, first time, as the idea of a phone one sent her into meltdown. She has fm and CFS in common with you plus other things.
As regards private doctors, you would need to be quite seriously wealthy I think; I used to deliver for a pharmacist including rare trips to a private patient. I know his meds cost £100s 25 years ago, and consultations and letters don't come cheap either.
As soon as you are able to, I would suggest you check when your assessment is done, and when it is, after two days request a copy of the report. Being paper based does not in our experience stop the assessor leaving out things, twisting some and inventing others. You actually can write in to DWP if this happens to you, coldly taking it apart with reference for each point to your evidence. But you need to do so quickly to get to the Decision maker before they take the report as gospel.
Very best wishes, Denby