Same - have a few flats in London that I rent out, and we aren't even allowed to put parquet/laminate on the floor if we renovate. The house owners association demands moquette... which sucks, as after every tenant leaves it needs to be replaced as it gets filthy immediately (even in the bathroom ).
Balcony nets, mesh, parabolas, or anything that can affect the visuals of the building all need permission...
Did you miss this bit in the original post?
And the fact that ASLOCA said that small pets may include cats and may not.
Also, the "He" in this case . I.e. an employee of the landlord or his representatives. If taking the apartment rested on this written confirmation, then it can be construed as part of the contract. (I am not a lawyer - I understand Urs Max is.)
OP - keep the cat, put up the mesh, find a new apartment where they don't muck you about. You might also like to write to the employee's boss and very politely complain. My guess he screwed up and is trying to back out. Who knows? The boss might be more reasonable.
One last thing: I've seen cat meshes on balconies all over the place. They're not uncommon.