Rabbits are crepuscular - they are most active early in the morning and later at night. (around sunrise and sunset)
I agree - don't put a rabbit in with a chinchilla.
he rabbit would sort of live on the bottom shelves, and the chinchilla would have it's own space in the top.
What makes you think the rabbit would stay on the bottom? The rabbit would probably love the top area of the cage! Lots of people I know have rabbit cages with upper shelves for the rabbits to hop up to, and the rabbits love it!
Rabbits can be territorial and it's sometimes hard just to get two rabbits to get along. There is a huge possibility that it will just harass the chinchilla, maybe even hurt it. And I don't know much about chinchillas, but it could probably do some damage to the rabbit as well.
I think if you want to keep both, you can either build some kind of a stacked cage to save on space - like one cage on top one on bottom, but they are seperated. You could also litter train the rabbit and rabbit proof the room, and let it free roam instead of caging it.
If you can't seperated them, you should probably rehome the rabbit. Maybe if you tell your ex that if she leaves it with you, your just going to find a new home for it, then maybe she would change her mind and take it with her? On the other hand, if she's too lazy to care for it maybe it would be better for the rabbit if you just find it a good new home. You could also do a search for rabbit shelters in your area, and bring it there.