My lease states that all drain blockages are the responsibility of the tenants. I provide a bottle of the hair and grease drain cleaner rather than tenants using the other stuff that can damage the system and if they use the bottle I provide they replace it. If a plumber is called they pay for it.
IMO, you lost the option of choosing your own plumber once you involved the landlord.
Just out of curiosity, what kind of dog do you have that would cause a drain blockage with its fur?