For Clarity, RB can save in BB format, but you don't get all the RB 'stuff' in a BB file format. (MGU,SGU etc are in the Save As options)
What you are asking for doesn't work that way currently.

To be honest, it's weird to me to hear someone wants to go in that direction.
I use BiaB to quickly narrow down chords/styles.
Then I go to RB to start refining it.
Seems strange to me to want to go back the other way.

One thing you mentioned that would be a cool feature is to Save As a new style.
In your new style the bass may use Pop/Funk and the guitar use a Rock Rhythm and the drums use LA Smooth (or whatever) .. that would be pretty cool.

I suppose Save Special-Save As Template is close, but that Template doesn't show in the Style list .. an ability to create a new style in RB would be pretty cool. And save a couple steps in your request, plus be available in both BB and RB..

/I love Simon's signature

Make your sound your own!
.. I do not work here, but the benefits are still awesome