Breastfeeding and joint custody :(?

My husband and I are getting a divorce and he is filing for joint custody. My daughter is a year old and I know that she's old enough to be off the breast, I'm just having trouble getting her to drink milk. I have been trying to transition her for a few weeks now and she refuses anything but the breast. I'm serious, she'll dehydrate before she drinks from the sippy. I've gotten a syringe and an unused medicine dropper and given it to her that way but she just spits it out. I'd hate to put her on formula the nights she's with her dad but I can't get a darn thing when I pump. She only nurses 2 times during the day and then wakes up to comfort nurse during the night (I'm not sure if she even gets anything then or if she is just using it as a paci because I wake up with full breasts every morning). I really just freaking hate this, I wish this weren't our situation but it is and I'm feel guilty in so many ways and now I feel sick about ending breastfeeding sooner than I thought. Any advice on how I can get her to drink from the sippy? Should I just try formula? I can't see how that would be any different for her than cows milk in a sippy, but I'm all out of ideas. Thanks everyone.