We had experience of this with a family member. They had attended one consultant privately, and had treatment privately. Then had another consult and treatment publicly. Got a bill for private, queried it, and they changed it to Public. Then had a third treatment via a consultant they see in the public clinic. Had treatment, at all stages hospital staff were told they were public patient.
But accounts want to put it through as private on the insurance. We've spent 2 years arguing with them, as they insist they were a private patient. Even though they see the consultant as a public patient, the admission letter was to a public ward etc. They have no record of a request from the person for private treatment, and give a different reason each time why it was private. In the end the insurance was sent a letter explaining the situation and asked them to sort it out.
Maybe I'm wrong but it seems to me, that for all the players involved, consultant, hospital, insurer are all better off if its done privately. The only person that's worse off is the subscriber with increasing premiums. As such (in my experience) you're on your own getting these things corrected.
So my advice is get everything in confirmed in writing, and the names and details of everyone who is informed that's its a public treatment, not a private one.