I was worried with my 3 year old because she doesn't like milk very much. I thought as long as she was getting yogurt and cheese she was fine. Our pediatrician said that although they do have calcium, other dairy products are not a substitute for milk. They have Calcium they lose it when they poop without the Vitamin D that milk provides, so he said. Talk to your doctor to see how he feels about taking the milk away because ours was very clear that milk is necessary in toddlers. In fact he told me to put whatever (strawberry or chocolate flavoring, etc) I had to into it to get my oldest (3) to drink it. He said that not getting enough milk at a young age will cause osteoporosis when they are in their 20s.
We also have a 2 year old 'milk-aholic' at our house, but the doctor said she was fine unless she drinks more than the recommended amount per day (and I can't remember what he said that was). However, I know that she drinks at least 12 oz a day and the doctor said that was fine.
To get him to eat, try to make the meals fun by letting him help when possible (stirring, pouring ingredients, etc). Plus, I try and always have fruit on hand because my girls love it and I usually give it with breakfast & lunch. I would also really limit the amount of snacking he's doing. If he gets cookies after leaving an empty plate at dinner, that is not teaching him anything. Make cookies a dessert after meals when he's eaten well and give him (somewhat) healthy snacks during the day like fruit, veggies, crackers, applesauce, etc.
I hope something I've said helps. Please remember that no matter how well intentioned moms are by giving advice, your doctor always knows best so give him a call. I hope you get the answers you need soon.