I know nothing about PCP but can comment on how I got a private MRI if that helps?
Basically I did a search and several options came up near where I live (Manchester). The costs and waiting time seemed to vary considerably.
It seems you need a doctor's letter for an MRI, you can't just book one yourself. However, several private doctor's practices will do a telephone consultation and write a recommendation letter for about £30. Make sure this is done properly - mine was so poor it didn't even say which knee or give any of the history I told the doctor, and the radiologist therefore struggled a little with the lack of detail.
The same medical practice / agency (I'm not sure what to call them) then charged me £220 and booked the MRI. I had to wait about a week.
I could have paid less if I had been prepared to wait longer.
I know you said you want this privately, but given your history I would have thought you were eligible for an NHS MRI, even if you had the subsequent private treatment.