It is very important that your daughter get calcium, vitamin D, protein, and fat. Yes, FAT, at least until 2 years, they need fat for their nutrition to aid in proper growth and brain development. This is the reason the doctor wants her on the WHOLE milk.
I was lactose intolerant as a child, but didn't realize it. I too would throw up every time I was made to drink the whole milk, which was almost all they had then. My grandson was having some trouble w/milk and my dauhghter eliminated it from his diet BUT she SUBSTITUTED other beverages, like soy milk or rice milk and made sure they had the adequate calcium, protein, and fat in them.
My other daughter was allergic to milk which is more serious, she could not have any milk products, so we used a very expensive formula until she was 2 from a sippy cup. At your daughter's age most pediatricians are not willing to run a test for milk allergy and even less likely to test for lactose intolerance, as it is hard to detect at this young age, 3 years is average. There are blood tests they can run, if she ALWAYS throws up when drinking ANY milk.
You might talk this over with a call to your pediatrician's nurse and mention what is happening. My guess is they will have you keep a food diary of what she is eating, because it may NOT be the milk, it can be the combination of milk w/another food. Do this by process of elimination. Only introduce one suspected food a day for 4 days, once she tolerates it, you can add another, each time you've eliminated one, you will come closer to the problem. I know this sounds like a drawn out process, but it is the health of your child and her health is more important!
About me:
I have extensive medical experience as a nursing major and I’m going back for my Physician's Assistant degree. I've already had many of the courses for the PA program. I have quite a bit of "hands-on" experience raising a daughter with multiple medical issues and later taking care of my Mom her last year of life due to incurable cancer. I was her nurse 24/7. I cannot tell you what to do, as I don't have my license, but I can make suggestions.