I think my daughter should go live with her dad

Me and my daughters father were teen parents. Neither of us were ready for a child. However, i accepted my responsibility more steadfastly than him, probably because i was the one walking around the school with the big belly. Fast foward 15 years and in the past 3 years my daughters father has been very active in her life. He is always there to help in every way accept financially. He has 4 kids, 4 different baby mothers but doesn't have a job. He makes small money from a hobby of his. I personally don't think that is fair because he needs steady income to support his children. I never say anything about thos though because he is a good father to our daughter. However now that my daughter is a teenager she has become increasingly disrespectful and manipulative. Its basically to the point where if I tell her to do something she doesn't want to at my house, she just huffs and puffs and goes to her dads house till whenever she feels like she wants to come back. She lies to her father and says i dont care about her feelings or im always angry at her. Which is the farthest thing from the truth. Her father says well he doesn't have any problems with her when she's with him. I always talk to him when im having issues with her because i feel thats the best way to coparent. (Backing eachother up) The truth recently came out that he has not been doing that. He feels that im exaggerating when i tell him how bad she has become. So pretty much at this point she feels like she doesn't have to follow any of my house rules and she can do whatever she wants because she can just call her dad to come get her and then just return to my house whenever she feels like it with no apology or repercussions. I feel like im a prisoner to her in my own home. I feel like i have no choice but to tell her to go live with her father. I beg her to talk to me but she wont. She rathers to talk to her dad, but come to me when she needs money. I feel like this is a lost cause because unfortunately its a cycle i cant break if her dad continues to not support any discipline i try to give her. I feel i should send her to stay with him so he can get a clearer picture of how she is daily and not just from time to time. Also i feel like staying with him for a while will teach her to be appreciative for how hard ive worked for all these yearsto give her everything. Again, her father took a while longer to mature than i did so hes only been active like this in her life the past 3-4 years. I want to send her to him, but i ask myself does that make me a terrible mom? I could understand a son, but who sends their daughter to a father?? Im ashamed but i can't continue to live like this. Please be kind with responses. Im 6 weeks pregnant and at work right now with tears in my eyes