We were originally turned down when we requested an assessment. After finding out their reasons for refusal, we then spent 6 weeks collecting the evidence to address there concerns.
Like you we had reports from various professionals, and also provided a log of incidents at the school, instances of where our son was unable to access the curriculum or socialization at lunch, specific concerns raised by the school, measures implemented already and the impact of these, and the challenges which we believed needed to be addressed.
Based on our experience, I wish we had submitted all the incidents and reports earlier as it would have saved us a lot of stress, worrying and running around as we tried to build a case of evidence to address their reason for refusing to do an assessment.
We also were in constant constant with the Ed Psych asking for advice on strategies to use when my son was struggling in school, which meant she had greater visibility of my Son's challenges when she was called on to write her report later on.
We have just received our EHC, (originally with 30 hours one-to-one support a week in a mainstream school bu now in the process of being adjusted to specify an ARP placement).
At no point have we had to appeal or go to tribunal - I do think this is because of all the evidence gathered and submitted.
Best of luck with the EHC - I do hope it goes smoothly for you as it can be quite a roller coaster ride for the whole family.