Aguu, I am so sorry that you are even having to ask this question. I have been through what you are going through...we had two pets, a yellow Lab, who was my husbands dog before we got married, and Mimi, our cat, who came to us while we were married. Somehow, we just knew that Chuck should stay with "Dad", and Mimi came with me. We didn't have any arguments over it, but I continued to love Chuck (still have his picture right here on my desk), and Mimi is still loved by her "Dad", but lives with us. Chuck went to the Rainbow Bridge a year or so ago at the age of 15, and thank goodness, Mimi is still going strong at 12 years old.
I don't know if there is such thing as "joint" custody for a cat, but I have several friends who have shared custody of their dogs.
Good luck to you. Hope you don't have to make a judge decide this, but if you do, sounds like you should come out on top.
Logan