My 2 1/2 year old has a dairy sensitivity, but it's not lactose intolerance. She used to get rashes on her face or neck (anywhere a dairy product touched her skin when she was eating or drinking) and then get hives on her torso within the next hour. Her pediatrician tested her for dairy allergy, and we were told she is not fully allergic and would likely grow out of the sensitivity over time. A couple of times after she turned two, we let her have dairy (mac and cheese, some yogurt), and she's thrown up both times. Just like you said--no fever or sign of illness, and totally happy and fine after throwing it all up.
She's been drinking rice or soy milk since she was one, eats non-dairy cheese, soy yogurt, etc. The rice and soy milks she drinks are all fortified with calcium and other vitamins, but they tend to be pretty low-fat. Since she can't drink whole milk and healthy fats are very important for brain development in little ones, we give her lots of avocados, which she loves. (This is what our pediatrician recommended.)
Good luck!