An Ed psych cannot diagnose Autism. The diagnosis needs to be made by 2 professionals. They can be any combination of the following: clinical psychologist and speech and language therapist and occupational therapist. This is the case whether done through CAMHS or privately.
If your student does have such a diagnosis. And you have the correct report, then there is no need for reassessment. You apply through AAO without a form 8. You will still need teacher evidence and support and you will need a detailed file note for the student’s file.
Letters from GPS are never acceptable under any circumstances. Any letter would need to come from a specialist or consultant.
I hope this helps.