IMO when you H chose to become wayward he chose to leave you AND cat behind. I would not suggest shared custody.
Definitely not best for the cat, but also seems like you are perhaps reaching for a reason to guarantee you'd have continued contact with H moving forward?
Me:36 W:30 M:2.75 T:7 BD: 4/2015 ILYBNILWY: 5/2015 W Moved Out: 5/2015 W filed for D: 7/2015