Big Cat Public Safety Act This bill amends the Lacey Act Amendments of 1981 to prohibit any person from breeding or possessing prohibited wildlife species (i.e., any live species of lion, tiger, leopard, cheetah, jaguar, or cougar or any hybrid of such species). Breeding means facilitating propagation or reproduction (whether intentionally or negligently), or failing to prevent propagation or reproduction. The bill revises the list of entities that are exempt from Lacey Act prohibitions regarding those wildlife species to include: (1) certain entities that hold Class C licenses in good standing under the Animal Welfare Act, and (2) current owners of animals that were born before this bill's enactment if the animals are registered with the U.S. Fish and Wildlife Service. Those entities and owners may not allow direct contact between the public and the prohibited wildlife species. A person who knowingly violates the prohibition must be fined not more than $20,000, or imprisoned for not more than five years, or both. The bill extends forfeiture provisions to fish, wildlife, or plants that are bred or possessed.