I discovered my baby girl has a milk intolerance or sensitivity when she was about 1 month old (she's 4 months old now). I am breastfeeding and wanted to continue to breastfeed so I put myself on a rigid dairy free diet to help her. It took a couple of weeks for it to clear out of both our systems but she's much much better now. She'd been having terrible eczema, diarrhea, diaper rashes, and gas that would rival a football player. When I say terrible eczema, her face and eyelids were swollen with it. All of it cleared up now.
I would recommend taking him completely off milk and see how he responds. If there is no change, then you can add it back in. Just be aware of the foods that have hidden milk in them (if you are going to do it, do it right, you know!). Read labels on everything and avoid anything with milk, casein, and whey. You don't mention if it's lactose intolerance or milk protein allergy but eczema usually points towards the milk protein allergy. Sucralose (artificial sweetener Splenda) is in a lot of stuff and has dairy in it. So that means avoid drinks, foods, or gum with sucralose. You also need to watch for sausage and deli meats. Sometimes they use milk as a binder in those foods. Same thing with instant oatmeal, there is milk in some instant oatmeals (depends on the flavor). If you give him vitamins, read the label, because milk is used as a binder in some vitamins. You also have to watch out when you go out to eat because you never know when a restaurant cooks something with butter. I just wanted to share what I learned because I was surprised about some of them. I found alternatives to all this stuff in Whole Foods or the organic section of a conventional grocery store.
I know it sounds overwhelming and a lot of work, and it is for about a week or so. But you'll find a groove for him and it will be second nature. For me, I have very mild allergies to citrus foods. So I can eat them once in awhile and be okay. But I can't eat citrus everyday because it makes the allergy stronger the more I expose myself to it. I am worried that will happen to my daughter which is why I have been so rigid until we can have her tested for actual allergies.
Good luck friend! I have a great lactation consultant/nurse in my pedi's office and she was really helpful during my quest for dairy free diet so feel free to message me with any questions.