Cat not allowed, reason not valid

Hello,
I have a ground floor leasehold flat, in a block where there are 4 flats.
The lease states no pets without written permission from the freeholder but a single cat or dog might be considered. So I have asked for permission, and was told that I cannot have a cat, as there is a shared entrance.

However, as I said, I am in the ground floor flat, which had a bedroom converted and its window was converted to a back entrance, this is what I use. Therefore I do have my own entrance, from which I am straight in my own flat, I never use the shared one. I told them this, but they seem to ignore it. Also, my previous flat was a leasehold as well, with the same freeholder (the Council), and almost everybody in the block, and the other blocks were keeping pets, especially, dogs. So did my next door neighbour, and he did use the shared entrance, so how is it a valid reason for the refusal of my cat? not to mention keeping a house cat. It wouldn't have to do anything with the shared entrance (which I"m not using myself, not to mention the cat).

Is there any legal way that I can take up if I do want to keep a cat?