I never had this problem because I gave my son sippy cups as soon as he was able to hold something in his hand. I will give you suggestions anyways. If your worried about her not getting enough fluid don't because she may be stuborn but if she really gets thirsty then she will drink what is given to her. Two ways to go and the first is give her the sippy cup with her favorite drink in it and the bottle with something she probably won't like (don't use water because you don't want that to be a punishment) like prune juice, cranberry juice, lemonade with no sugar. She will eventually go ahead and drink the sippy cup but you have to let her see that your putting her favorite drink in that oe. They do know the difference. Of course I think this one is a drawn out, long, and emotionally tiring thing to do.
The other is what your doctor told you to do and just take it cold turkey. She has to eventually drink something and will eventually pick up that sippy cup and realize that it's not so bad. It will be a battle at first but in the long run you won't have to worry about the rotting teeth because of prolonged use of the bottle. Just keep reminding yourself that and you will be able to endure.