Offering a trial period isn’t a refusal – it’s a conditional approval. If the cat isn’t a problem, then any subsequent refusal would certainly be unreasonable.
That said, there’s something about this that makes me think someone is trying to keep everyone happy.
If you have intimated that you are only looking after the cat for a while, then clearly the “trial period” is a way of allowing you to do that without upsetting the neighbour who raised the complaint.
My advice would be to forget the letter of the by-laws and be the reasonable person in this. Accept the trial period. In any case, there is nothing absolute in giving permission for a pet. If the animal breaches other by-laws relating to noise and nuisance, or if it endangers the health of another resident, the owner can still be ordered to remove it.
If it doesn’t breach those by-laws, then they have no reason for not declaring the ‘trial” a success
Go with the flow and deal with the end of the trial period when it comes.
And yes, ask to see the details of the complaint for the simple reason that you want to be responsible owners and modify the cat’s behaviour accordingly.