[Keith]

That's stupid. It really should be Big Brother's Big Mouth.

Bare in mind who the previous host of Big Mouth was.

With current allegations involving both him and that programme being mentioned in Despatches, it's fairly easy to see why they've opted not to use that name for the spin off show. (Especially as there's the potential for further allegations to be made in the coming weeks & months, which might be directly/indirectly related to that programme.)