The bulldog's name is "Butch". He's sitting on a miniature puff box and is shown on p. 283 of Tom Felt's "L.E. Smith Glass Company--The First One Hundred Years". He also came in green and frosted crystal.
He's also shown on a different puff box in the Whitmyers' "Bedroom & Bathroom Glassware of the Depression Era," p. 12-13.