My kids are absolutely worse at home than school. They are better in both environments on the diet...

And I strongly disagree with Polarbear about simplifying being enough. There are some foods where that may be suitable. But if you are looking for food chemical intolerance like salicylate and amine you have to know exactly what you are doing to be able find the pattern and simplification won't do it. Allergy leads to fairly immediate response so the simplification approach can work for allergy (as does allergy testing in most cases), intolerance is a load issue, the reaction may not be 20 mins after eating, it may be 3 days after eating an increased load of the chemical in question through a pattern of foods you are not aware all contain the same thing. We understand that if we've had a beer, a wine and a vodka that we have had 3 drinks with cumulative effect. That just one of those might have fairly minimal effect and that three is quite likely to have a noticable effect. We also know that some people are effected much faster than others by alcohol, but that everyone has a point at which they are overtly impacted. What people don't tend to be aware of is that cheese, chocolate and oranges all contain amines and someone with an amine problem might be fairly ok eating one of them but be extremely depressed (or badly behaved) if they eat all three.

Also, intolerance may not come with obvious tummy upset.

Also removing one thing at a time you risk the possibility that symptoms from a food still in your diet will be masking improvement from something that is an issue but is not the only issue or is not the worst issue.

The first two weeks of an elimination diet is absolutely hellish. If it works you won't care and will be very motivated to continue. If it doesn't work it will be over fast and you'll never have to go there again.