What national department would be used to regulate pet laws and such?

Answer 1

IIRC, the Animal and Plant Health Inspection Service, part of USDA, has federal jurisdiction over inhumane treatment of animals, but that's usually in reference to farm animals, animal imports and animal diseases.

But laws regarding pet welfare are usually not handled at the federal level.

Answer 2

Usually pet laws are a local or county law or possibly even a state law. The USDA is often in charge of regulating 'farm' type pet operations (puppy mills) and so forth.