@OP
From the legal point of view, in general in the contract (Mietvertrag) it is detailed, that the building standard sheet (Hausordnung) is integral part of the contract. If this is the case with you I think you are generally in severe disadvantage in any legal dispute. Since you should have taken notice of that when signing the contract, respectively should have requested that Hausordnung for your information before you signed. Although the behavior of the landlord is/was not correct, he could just take any excuse to justify his misconduct pretending he confused things etc.. I highly doubt you will get anywhere in a legal course, since agreeing to the Hausordnung very likely has most weight here.

Most promising IMO is to try to reach an separate consensus with the landlord allowing to hold a dog, pinpointing on his original promies, and put it in paper.