According to an episode of Deep In Bear Country (timestamp 12:25), a fan podcast, Mama and Papa used to go by "Missy" and "Junior":
!"Of course, they weren't 'Mama' and 'Papa' then—just 'Missy' and 'Junior'"
It's from a book that was written by Mike Berenstain, probably The Berenstain Bears' Big Family Album because the plot matches. The podcast also says that Papa's father, Grizzly Gramps, is named Earnest (so Papa's real name is probably also Earnest) but I don't know what book that is from.