Yes unfortunately these days I find people place way too much value on cows milk for kids. My son never had any because it made him breakout in horrible rashes on his face (this when he was breastfed and I consumed dairy). I cut out dairy and voila the rashes disappeared. This intolerance continued until he was about two at which point it seemed he outgrew it, although to this day he is not a fan of dairy.

When I was a teenager I started suffering from pretty bad acne and it all again disappeared when I cut out dairy, sugar and caffeine.

All to say that dairy CAN be a good supplement to a child's diet IF they can handle it and only as a SUPPLEMENT, it shouldn't be their main source of calories and should only be a small part of their diet.

If I were you, I would just keep pushing the issue like I did even though our situations were not exactly the same. I feel your frustration!