How do you split pet care across two homes without stressing the pet (or the co-parents)?

Looking for advice from anyone who has actually made a shared pet situation work.

Quick background: I co-parent a teen with my ex and we have a new baby living with me. Our teen is really attached to the family cat. The cat is an adult, healthy, indoor-only, and has always lived with me. My ex just moved into a place that allows pets and wants the cat to stay with him during his parenting time so our teen can see the cat there.

I get why my ex and our kid want this, but I worry about the cat getting stressed by moving back and forth. When our house gets loud with the baby crying or the teen having friends over, the cat hides more. He hates carriers and car rides, and he is not a travel cat. There are also practical things to think about: litter box setup, scratching posts, keeping the same food, and making sure meds are not forgotten if we need them.

My questions:

1. Is it generally better for a cat to have one stable home even if the kid misses them sometimes?

2. If people do split time, what schedules have actually worked in real life? Weekly, monthly, only special occasions, something else?

3. Any specific tips to make transitions less stressful if we try splitting time? (Carrier alternatives, calming routines, keeping the same setup, etc.)

4. How do you balance what feels fair to the kid with what is best for the cat?

I am not trying to be difficult. I just do not want our cat to become something that gets tugged back and forth. Opinions welcome, especially from people who have done this long term.