Welcome to the site RobBob,
The two videos that you're referring to do not correspond with each other. The guitar lesson was done in the "early" Banjo Ben days (about a year before the mandolin lesson). Alot of the lessons Ben's teaching today are in relation with each other and are in the same key.
So, to answer you're question, yes you would be in the wrong key. It's a simple fix though. Since the guitar is usually doing the capoing, simply move it down to the second fret and you will now be in the key of A and still playing the exact same lick, just moved down one fret (or position).
You won't see the third fret capo position a whole lot in bluegrass, usually it will be the second or fourth or no capo at all. However there are times like this when you will be on an odd fret. After watching both of the videos, I personally like this song in Bb better. It just sounds right, but it takes a really good mandolin player to pull that off and Penny did it well. Most likely, Ben taught the mandolin lesson in the key of A because he understands that most of us here are students still learning and the Bb version would be much more challenging. Not only that, at most jam sessions, people will be playing in A or G.
Hope this helps.