There was no bird and it was never Berenstein.
People are often curious about the spelling of “Berenstain,” a phenomenon that’s much older than the Bears themselves. As Stan Berenstain recalled in Down a Sunny Dirty Road, the 2002 autobiography he co-wrote with wife Jan, even his fourth grade teacher had questions:
“On the very first morning, when [Miss McKinney] called the roll, she took exception to my name. She said there was no such name as Berenstain. The name, as everyone knew, was Bernstein—and that was what my name would be, at least in her room. When I raised my hand and protested that Berenstain had always been my name, she silenced me with an icy stare and said she didn’t approve of people who changed their names.”
“Berenstain,” it seems, is less common than other, similar variants. But there’s a simple explanation. According to family lore, the spelling results from an immigration officer’s attempt to record phonetically an accented version of the traditional Jewish name “Bernstein” as pronounced by Stan Berenstain’s grandfather. He had come to America from Ukraine, where the name would have sounded something like “Ber’nsheytn.” Since then, the family has always spelled it Berenstain, as it was originally documented.”
–From the Official Berenstain Bears Blog
So yes. The truth IS out there. [ Obligatory X-files reference]
Berenstain Bears Bibliography & Blog 
How do you explain even the US Patents and Trademarks Office *Appeals Court* getting your last name wrong in an official case in late 2001? I mean, such a grievous mistake would certainly bring into doubt the entire ruling on that matter. Courts do *not* make such typo mistakes ever. http://www.plainsite.org/dockets/o6juzaq6/uspto-trademark-trial-and-appeal-board/stanley-berenstein-and-jan-berenstein-v-kid-kore-international-ltd/