Hi
I would think the simplest answer is no.as there two different species. As chinchillas need a very different cage to rabbits. Chinchillas like to jump and climb and there also nocturnal. Rabbits do not like height on ledges like chinchillas do. Also chinchilla need a tall cage full of shelves, dust bath, hide etc. They rarely ever go to the floor. But what would concern me would be falling objects on the rabbit. Rabbits however are crepuscular (active at dawn and dusk) and would require more room from the bottom of the chinchilla cage. You could however try a diy cage and try building one from C&C grids? I’ve known pleople to do that.
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But your best bet is to bond them outside of the cage though your chinchilla will most likely be asleep most of the time!
there fur is soooo fluffy and soft ! I used to keep one as part of my animal management course he was so cool! Be careful handling chinchillas they jump can be easily dropped
Edit: I forgot to add this when I first posted! No your rabbit and chinchilla will not breed as they belong to two separate taxonomy groups rabbits are in the order of Lagomorpha and chinchillas are rodentia. Fun Fact! Only species in the same order can interbreed with each other. Isn’t zoology great?
Hope I helped