There is a scientific research, (no time now to search it for just 1 person)
It documented how babies did not react to allergy test because test is just a prick on the skin or mix of blood. 
But when the diet was corrected, change occurred.

There may be other causes for eczema so personal results may vary.

Ideally should test for 2 to 4 weeks abstain, then introduce the sensitive ingredient again, to validate findings.
But that would be more suffering for the child.

While I have benefit from some good doctors, I have been a victim of many doctors too.
Medical centers profit from patient visits, test and sales of pharmaceutical and equipment.
Some doctors are not even up-to-date. As Jack Ma said, experts are familiar with the past.

When you post on public forum like LYN, this is listening to public.
Only Laitlee and me recognize lactose intolerance. Intelligent people are a minority.

Edit: Animals' milk is different from human breast milk. For a calf to grow from infant to adult weight, what do you think cow's milk is made for? Not suitable for human? But if you disagree, why not buy a carton of cow's milk and drink it all yourself for a whole day. You won't die but your body will suffer the consequences.

Anyway, I am not profiting from you.
Did not sell you another product. Merely suggest you abstain, see the effect in 2 weeks.