Only, it didn't happen that way. They did countless one-shot cartoons with different characters, all created with the hopes of becoming Paramount's next cartoon star, but none of them stuck. There were recurrers such as The Cat, Swifty & Shorty, Honey Halfwitch, and Goodie the Gremlin (a Casper-expy), but none of them made any impact.
One of the characters they tried was Boobie Baboon, who managed to appear in two cartoons (both 1965) under the creative supervision of Howard Post. Oddly enough, both cartoons involve prisons (in one he's trying to get in, in another he's trying to get out). I wonder what other ideas they had in store had the character continued to appear in more cartoons.
You can judge for yourself. Here are the two shorts.
EDIT: According to Mike Kazaleh, Howard Post never used Eddie Lawrence in his cartoons. The voices are Bradley Bolke and Bob McFadden