First, get the schools behaviour policy. Read it carefully. They should detail what actions they take to deal with unwanted behaviour.
Then decide if you think they have followed all their own strategies first, exclusion should be a very last resort.
I found that DS was being excluded without other things being done first. They wouldn't even do it officially, just called it a 'send home', which is actually illegal.
If they are following all their proceedures, and he still needs to be excluded, please make sure it is done officially, with correct records being kept, and a letter to you each time.
This will help you to build a case for your DS needing more help to cope with school, and help if you do decide to apply for an EHCP.
Also, if you think a part time timetable would suit him, suggest it, but please think about it very carefully first.
My DS was put part time in October of year R, and hasn't done a full day since (he is now year 1)
Seeing as he had only been full time at school for 2 weeks before this happened, he has practically never had a full school day in his life. which I find very sad, and not how a child should begin their education. This is obvious in the way that he is now falling behind academically, adding to his stress at school, and meaning he needs even more 1-1 input to teach him the things he is missing in class.
I hope the paed appointment is useful, and pleased you have been able to use private healthcare to get it quicker.