This issue came up at mediation for me as well. I find it highly offensive that the ex boots me out of her life then expects to know who I associate with/live with in my new life when she is the one who forced me to undertake the new life in the first place!
If ever there was a time to use capital F off, then it is in this situation! If it came to court, I was prepared to say to the judge that I refuse to disclose such information. Is the judge going to withhold custody as a result? Perhaps, I would say fine.
(I do have some human dignity left)