Everything takes time. My 6 year old son sees a dietician from Childrens' Hospital in Seattle. He had a stomach tube, couldn't eat when he was a baby-he has a long medical history that involved him being on a life support machine, couldn't drink out of a bottle, couldn't suck, etc. He's fine now. But when he was learning to eat and drink, he wouldn't touch milk. He ate alot of yogurt, string cheese, different cheeses. When the dietician asked me how much milk he drinks, I told her he doesn't like milk and re-evaluated his list of what he does eat (a list that I provided for her). She saw that I had yogurt and cheeses on there, and she said, "oh, he's okay with his calcium because he's eating yogurt and cheeses". Yogurt, cheeses, cottage cheese, sour cream, cream cheese, these are all dairy products that have calcuim. Don't worry about him not drinking milk in a tippy cup, just make sure he eats more yogurt and cheese (with some juice so he doesn't get constipated). I'm sure you have a great Doctor, but if a Dietician from Childrens Hospital (someone who specializes and has a degree in Nutrition) told me yogurt and cheeses was fine, then don't worry about the milk right now. He is transitioning from a bottle to no bottle, and that is a big change for a little guy.
Also, I've learned-when you are anxious for him to do something like eating or drinking something, he won't want to because he feels your anxiety about it, and in an almost rebellion way (your son being the rebel), won't eat or drink it. This happened to us, I was so desperate for him to eat and drink. After 6 months, I decided to sit him in his high chair, give him the food, snacks, or whatever, and then walk away (to the kitchen, doing dishes or something). I kinda let him do his thing, as I was doing mine, and then he just ate or drank.
Let him drink whatever from his cup. My 6 year old now loves milk, mostly plain, but loves his chocolate milk too. Another thing that worked for us, was I used to put a tiny bit of chocolate syrup in his milk, and he would drink it.
Be patient, Good Luck!