Hi K.,
What a stressful situation for you.
This makes me think about what it was like for me when I was a teenager. My parents are divorced, and at age 15 I was interested in working the situation to my advantage. My father wasn't interested in being a dad and he was an alcoholic that certainly enjoyed his bachelor life. However, the potential to live with him was a new environment with more freedom, which is appealing to a 15 year old trying to assert some independence. I hate to admit it, but on a few occasions I did suggest the idea of living with my dad. Looking back on it, not only would that not have been a productive experience for me, but deep down I didn't want to live with my dad.
I was probably holding it over my mom to try to leverage something I might have wanted.
I think my mom called my bluff once and suggested we try it, and I backed down. Not sure you'd get that response from your daughter, your situation makes me wonder if your daughter is more focused on pitting the two of you against each other to get what she wants.
In my situation, my mom sat me down and talked through with me why I might want to move, and what that would mean long-term. She did this from as neutral a place as she could, explaining to me that if I really wanted to go, she would not stand in my way. Working through the logic, weighing the pros and cons, and understanding the impact helped me to realize that living with my dad wasn't a smart move.
I agree with what others have posted here. It seems like a good idea to meet with a counselor that can serve as a neutral party for you and your daughter to express your feelings on this. A move would be a big change, and based on what you have shared about your ex, he won't be focused on her best interest. This has potential to have long-term negative effects on your daughter.
Working with a counselor gives your daughter a chance to think through those pros and cons, and figure out the real reason for the move before making the move. This takes you out of the equation, in the event your daughter thinks you are biased.
Good luck with this!