LOL, ellen is spot on, I will go back to my paperwork, you dont need me 
Joking aside though, I had to be really persistant to get Dd3 a dx, she was 5.5 when I started the process and 9 when she got her dx of ASD, we tend to say she has aspergers because she fits that profile.
Dd3 has advanced skills in some areas and that what was the school was focusing on but she has what they call a spikey profile, that means she is really good at somethings and really poor at others.
My Dd3 has never ever been able to do puzzles due to her poor coordination but a LO I am currently working with who has ASD is amazing at puzzles but is totally imersed in them at the expense of everything else.
I guess that the school are like Dd3's old school, because she didnt melt down she was fine. The fact the she used to come out of school looking grey and used to run off out of the gates obviously didnt register with them
This is your chance for a fresh start but you need to go and meet the Heads and Sencos. Ultimately you will probably need their support to get a dx becasue the proffs are very reluctant to dx without another so called professional opinion.
Dd3 had a speech and language assessment which was invaluable because it highlighted that although she has very good language her communication skills are poor, she only talks about what she wants to talk about and is not able to conversation share. The SALT picked this up very quickly and also the fact that she was very literal in her language.
TBH, the very best bit of advice I had when she was being assessed was to keep a diary. I recorded and dated all incidents of unusual/quirky behaviour, what caused it [if I knew] and how I dealt with it. The proffs really sat up and took notice when I took it too them.
I think you should go back to the paed if you havent seen him/her since the EP saw her because you are still concerned and nothing has changed.
Good luck