Help with getting a referral for diagnosis, no support from school

Our son is 12 years old and about a year ago he read a book written by Libby Scott, a teenager with autism. He came to me saying that he thought he could be autistic. We have always been aware of his quirks and I believe he probably is autistic. After lots of discussion about whether or not to look for a diagnosis, we decided that a diagnosis would be the right thing for our son.

I spoke with my GP who advised we'd need three months observations from his school before a referral to the umbrella pathway could be made.   

I spoke with the school's SENCO and she said that although she could see some of the characteristics, this doesn't present a barrier to his learning and therefore they wouldn't recommend a referral.

I then went back to the GP who said they couldn't refer without a report from the school.

We have been left in a position of not knowing how to move forward from this.  We have just paid for a private initial assessment and the outcome was that they recommend a full assessment for our son. We're not in a position to pay for this privately and would need to proceed through the NHS. I am speaking with a different GP this week to see if they will refer him with the psychologist's report rather than something from his school.

I would really like your thoughts or similar experiences of difficulties getting referred. We have worked so hard over the years to support our son and we have put in place lots of coping strategies so on the surface it may seem like he is coping but he actually finds lots of everyday things challenging and stressful.  He feels that getting a full assessment and finding out whether or not he is autistic will really help him understand himself.

Thanks in advance for any advice.