After many more hours than I'm proud to admit I've come to the conclusion that for some reason around 2002 things changed and people started using Berenstain exclusively. I have found examples after 2002 although generally they are college papers/amateur posts that lack press releases, strict research etc. I'm unable to find any professionally printed archived articles or files with Berenstein after this year.
I have however found an article using both within one sentence... maybe this was in between 'the switch'. It references the Bears as Berenstein yet the authors have their last name as Berenstain. You can view it below. If you're able to find anything after the year 2002 using Berenstein please use the contact page. Common sense would tell us that things after 2002 might be too new to archive and scan... or that articles after 2002 were more commonly posted on the internet instead of on paper. The 2002 theory seems more interesting.