Lots of schools now use private EPs for various reasons. Sometimes an LA will use this as a reason to reject but it would stand up in court as they are just as professional as LA EPs . The same is true of OTs, speech therapists etc.
For example, I know of a pupil whose parents requested a place at a very expensive independent special school. The LA refused. The parents had reports from private EPs and medical therapists to discuss what therapies were needed. They won the tribunal and in fact the judge additionally ordered several hours a month of respite to be added to the EHCP.