Could a cat get sick or die from eating a poisoned mouse?

Three times now, I’ve found a dead mouse in our basement. Already dead, not killed by a trap. (We’ve also seen one live one and caught another in a trap) We live behind a restaurant, so I suspect that mice are eating some kind of poison that the restaurant uses, getting into our basement, and then dying.

We’re thinking about asking our landlord to allow us to get a cat because of the mice. However, I don’t want to potentially secondhand poison a cat. So, my questions:

1. Is my theory (that the mice are already dying of poison and just die in our basement instead of elsewhere) plausible, or could there be another reason that a mouse would die in our basement like that?

2. If the mice are being poisoned, how dangerous would it be for a cat to catch them and/or eat them?