My friend had to agree to a code of conduct for his dogs as part of the CTTT ruling. He had already provided his EC with such a statement but they wanted to take it further, mainly on the urging of just one member.
As part of your preparation to take this further, get all the information about your cat prepared as is recommended by the websites I mentioned (their vaccinations, letter from vet) and that they will not be free to wander on common property, will not pee in the corridor nor claw the carpet in the foyer. And that all waste will be properly disposed of (wrapped and placed in bins).
It would be reasonable to deny the keeping of a pet if such pet caused problems in the complex and the owners did nothing to address them. It is unreasonable to deny the pet simply because it resides there unless the bylaws state no pets. And I would be inclined to bring the cat back too.