Landlord Asking for Extra Rent for Pet in an Unfurnished Flat

Our lease doesn’t mention pets, but now the landlord wants extra rent for keeping one. (we have a small kitten 🐈) We’re fine with a pet fee, but their reasoning is “cleaning and damages” * The flat is **unfurnished**, so no landlord furniture to damage. * We might **move out in April**, so changing the lease for 2 months seems pointless. We offered to let them deduct any actual damages from our deposit and we will take care of deep cleaning when we leave, but they still insist on extra rent. what should we do in this case? any insights?