My daughter is lactose intolerant. She was when she was young, grew out of it, and now it's back. We had mashed potatoes the other night made with soy milk -- disgusting. When she was two, we used to pretreat her milk with lactaid which reduces the lactose content in milk by 99 percent or so. It's expensive but well worth it if you like your milk. That was mainly because any other substitute would end up being hurled across the room.
It also seems to be related to amount. Small quantities of cheese and yogurt do not have much of an effect, larger ones do. Ice-cream in any amount bothers her and so does any amount of milk. We have started buying her sorbet which she likes instead of ice-cream.