OP ask yourself the question what you would have needed as a 15 year old should that have been your scenario?
Your 15 year old clearly needs some guidance from the adults in his life, yet his own mum was unable, for whatever reasons, to cope with him, sending him clear message he is too much. Rather than perhaps parent him?
Its easy to judge on the internet when one has not got the full picture.
But at 15 you are not only stubborn, but feel you know it all, can do it better, sleep in, ignore piled up rubbish but can also be the best company, and loving.
Your son needs all the adults in his life to care for him., hear him out, and guide him. That is it. He did not ask to be put in the position where his family divorced, he then had to leave what he knows at home because mum was unable to come, and live in another home, a lot to ask from someone who is still a child. It is huge in fact. Regardless of how well they may cope on the outside to everyone else, this is hard for any child. He did not ask for any of this.
OP, you also seem to think this is someone elses problem by saying your current wife is unable to live under the same roof as your son. What message is that leaving for him and your other children? He is not a kitten you can take to a shelter once you have had enough.
Is your son causing issues?
And if so what issues?
Is there substance misuse?
Depression?
Get to know your son and his needs by having a genuine chat, not just 15 minutes, give him the opportunity to have a voice, then from that all adults need to come together and work together.
Would you feel it was fear if you got asked to leave home because you called your DF a bitch? Or does she genuinely feel unsafe due to other incidents?
Or does she simply use this incident as an excuse to keep home for just the 3 of you?