Food Allergies Treatment

She is 3. She has severe skin reaction to egg white and suspected intolerance to wheat. She had eczema as a baby but we took her off milk and it cleared up. My GP was difficult in prescribing soya formula for her but eventually gave in.
Me and my partner have reasons for refusing to have her tested at the local NHS hospital.
We looked into going private for allergy testing and found a pediatric allergy specialist in south east London. They said i needed a referal from my GP even if i get the testing done privately.
My GP refuses to refer her saying we should use the local NHS hospital.
What can i do … where do we go from here?
I can not change GP’s since they are all pretty useless in my town and the only ones that aren’t are not taking on new patients. (My last GP was brilliant but he retired from the NHS to join the police service :~(