Continuity mistake: When Fred and Barney arrive at the bowling alley, they are seen standing in the doorway, but the very next shot shows Fred inside the bowling alley and Barney is nowhere in sight.
Audio problem: At the bowling alley, there is a red haired man who is initially voiced by Alan Reed, but when he says "Hey guys! Did you get that?", he is voiced by Mel Blanc.
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