I think what you're trying to do is a good thing (help your child be healthy) but want you are inadvertantly doing is teaching your child not to listen to her body. Years ago, my second child wanted peanut butter but wouldn't touch it once it was in front of him. I trust his body to say no to things even if his mind is saying yes.
Picky eating is a classic sign of food issues. There is a yahoo group called foodlab that was very helpful to me when I first discovered dairy was an issue for my oldest. I'm still on it & it has been very helpful throughout the years. There are many, many on there who do not consume dairy. You said your/her dr. said she doesn't have a problem w/ milk. I wonder how he/she knows? That person is w/ your child, what, 15 minutes out of months, if that? Additionally, drs can be swayed by the "research" (what the dairy council spits out) as much as the next person.
The dairy council pays 100s of millions of dollars on ad campaigns designed to get us to buy more diary. They don't have an income if we stop buying dairy.
Mark Bittman recently wrote a couple of articles on dairy for the New York Times. I don't have links but if you go to NYT, & searched for him and/or dairy, I bet you'd get it quickly.
My children haven't drank dairy since my oldest was 1.5 years old & an only child (I know have three - days away from 10, 7 & 3). We get our calcium from the sources humans were meant to get it from. Humans weren't designed to drink the milk of other mammals, no matter what the dairy council says. What other mammals drink the milk of other species of mammals?
My midwife was just telling me a couple of weeks ago that she no longer recommends mothers take a calcium supplement & that the recommendations no longer say to supplement. She recommends food sources, whether that includes diary for individuals or not.
Now, in our case, I don't believe our systems are healthy & that we aren't absorbing all the nutrients we get via food (because of candida overgrowth & leaky guy) but that doens't mean we need dairy. It means we need to heal our guts (not a simple task, unfortunately).