Other mistake: When the Inventor asks the Flintstones and the Rubbles if they are game enough to try his Time Machine, Betty says, "Golly, I don't know." After Betty says the line, her nose disappears.
Other mistake: Fred's telephone cord is missing when he and Loudrock are calling their friends about the puppies.
Monster Fred - S5-E2
Other mistake: Fred's mouth doesn't open when he says "Now knock it off!" to Barney in his car.
Answer: The name of the pet is "Baby Puss." He has appeared in episodes, but only a very few. The first episode was probably "The X-Ray Story" from season 2.
Bishop73