Just had to post about my experience:
I had a 5yo in preschool who was not following rules, running away from the teachers, hitting, kicking, biting and more. I was asked to take her out of the school.

She also had HUGE outbursts. Over nothing. (Apparently.) However, once we received her diagnosis and started working with a behavioral specialist, we figured out that her outbursts related to "unexpected" situations.

Basically, if something unexpected happened, she would lose it. That was why it was SO DIFFICULT to pinpoint what was causing her outbursts. They could be about anything at all - the common thread was that they were unexpected things.
I took her to a private child psychologist and she was diagnosed with Asperger's Syndrome, and she has a massive IQ. She is now 7yo and reading on a high school level.
We tried diet elimination. To no avail.

Because what was going on with her wasn't really about diet, it was about how her brain was wired and that fact that she was so advanced that she was WAY, WAY ahead of everyone else.
What really helped her over the past two years was an intensive behavior plan, overseen by a behavioral specialist.

I'm curious what the paraprofessional does with your child all day at school?
In my DD's case, her paraprofessional was guided by a behavioral specialist who set up a very specific plan and approach to helping my DD learn how to behave at school. She had visual aids, wiggle breaks, special seating in the classroom, social skills groups, and more. She improved immensely.
I agree with the others that you NEED TO GET AN OUTSIDE EVALUATION. Do NOT rely on the DOE to diagnose your child. And if your psychologist says there is nothing wrong with your son, then go to another one until you get an answer.
Then, take that diagnosis to the school and get an IEP.
I LOVE Wrightlaw, too.

They came to my area and held a conference and it was amazing. Every parent with a special needs child should attend one.