Hey there
one of my twins has a problem with some dairy- from what we can tell its a protein in dairy that is the problem. It seem to flare up with whole fat dairy and is more tolerable with low fat or not fat dairy products. Its more of a malabsorbtion than an allergy. She coughs until she vomits for days and days after she has a malabsorbtion.
She can handle a certain amount of dairy, but her body gets to a point that it cant tollerate the protein that causes the problem any more, and she gets sick. It is hard to tell what food has more of the protein that she has trouble with, so we just stick to what works and challenge her every now and then.
Does this sound similar to your child?
It has gotten a bit better since it started- which was when i weened them at 12 months she got gastro and it stripped her gut- it has never been the same. As she has gotten older she can tolerate a bit more dairy- can even have a bit of cows milk every now and then now- but we still have to be careful.
I was fortunate enough to get onto a natropath that dx this issue- i had previously seen several docs, peads and specialists who could not give me a solution to this problem. The natropath gave me a bottle of homeopathic drops that work on tightening the filters that allow this protein to go through to the body, hence stop the vommitting.

btw to tell if a child is going to react to a certain food, find their resting pulse by taking their pulse for a few weeks whilst they are eating and nice and calm to see what their resting average is. Make sure during this time you are not giving and of the suspected allergens. After their resting heartbeat has been established, challenge them with a little of the suspected culprit. Whilst they are eating take their heartbeat at 10sec, 30sec, 1min, 2min, 3mins, 5mins , 10mins , 15 mins to see if their heartrate changes and becomes accellerated whilst eating this food. If it goes up, then the body is reacting to the food!

HTH and good luck!