Sorry, kinda long / Different perspective to think about
Our older dd is milk allergic (no, NOT lactose intolerant) and I first figured it out when she was 3-4 mos old. Her symptoms were extreme fussiness, "colic", a sandpaper like rash and a red anal ring rash that looked like a burn whenever I had a lot of dairy in my diet. When I mentioned it to one of our peds he told me it was a coincidence and tried to explain away the symptoms, but I countered with the fact that pre-mama days I was a molecular biology associate and knew how to conduct a trial thank-you-very-much! and we agreed to disagree. He did add that most milk "sensitive" kids outgrow it between 6 and 12 mos, and to try it later. Well, at 9 mos we caved at her plainitive lunges for a soft serve ice cream dh had, and
she got hives, swelling, and a flame red rash. She also gets 1-2 weeks of colitis (no blood thank God) if she ingests any milk proteins. Fast forward just under 4 yrs, and she is still milk allergic. We educated ourselves and we're doing fine, mostly.
If your child's symptoms are any of the "classic" ones like rash/hives/vomiting/breathing trouble please consider consulting a pediatric allergist before trials. Repeated exposures can increase sensitivity in food allergies, unlike other allergies which sometimes respond to desensitization. Although a lot of kids' milk allergies/sensitivities are mild and are relatively quickly outgrown, some are more serious and like my daughter's, progress to potentially life-threatening levels. We have had an EpiPen Jr for her since her diagnosis at 9 mos, and never had need for it until last week. Her otherwise very loving and doting gf forgot
about the milk allergy and put cows' milk on her cereal. Luckily, my dh was there and when she started to complain her tongue felt "spicy" he went into the kitchen to check on her and saw the hives. She responded well to a hefty dose of antihistamines, but if she hadn't quickly she would have gotten the Epi and a ambulance ride to the ED. I'm not trying to spook anyone, or elicit sympathy, just put some facts out there that someone reading this thread might need.