I am helping a boy who has just been diagnosed with learning disability and we are in the process of requesting an EHCP. If you choose to go private, a Clinical Psychologist will be able to give your son a formal diagnosis of learning disability. However, the SEN Panel tend to look at his progress at school rather than any diagnosis. They want to see evidence of support at school, which includes any interventions and strategies implemented. Diagnosis is important but evidence of progress and level of support in school are more crucial. This is an exception if the child has significant education needs that the local authority believes that the current school will not be able to fully support his needs.
From my experience, schools are not very keen on helping those have relatively ‘mild’ special education needs, as they have to prioritise their SEN students according to their severity. Therefore, you need to express your concerns and wishes to the school and push the school to get things done the way you want.
If you wish to get things moving more quickly, my advice would be acting both on the school and EHCP. While addressing your main concerns to the school, you can go private to get a diagnosis and then apply for EHCP by yourself. If the SEN panel want to see evidence at school, they are probably interested in seeing My Plan (informal plan made in school similar to EHCP). Do talk to the school about My Plan. It might be time to start it.
If you want to get more advice and support to aid you in the process, you should contact SENDIASS, who will be able to guide you in dealing with school issues and the EHCP process.
Hope this helps!