Is there a law that states the deposit must be one month? You could adjust the contract so that if a tenant requests pets, they're liable for a larger deposit / cleaning bill when they leave.

I rent and have two cats, they pee in the litter tray so that's not an issue, but they cause all sorts of other damage, mine like sitting on the bannister at the top of the stairs, so there's loads of scratch marks on it because it's too high to jump up to it easily, they're very furry Norwegian forest cats that groom each other a lot, so I wake up frequently to vomited up cat balls on carpets, etcetera.

My deal with the landlord is that I'll just pay for whatever damage is done, even if that's higher than my deposit.