A couple of ideas..

Go to Apple System Preferences | Security & Privacy | Privacy. Scroll download to Files & Folders and look for GarageBand - make sure that "removable volumes" is checked. Also, you should see Band-in-a-Box listed there OR in the Full Disk Access section.
After that, quit GarageBand and re-open.

In the Band-in-a-Box DAW Plugin, open the Preferences and make sure that the first three folder locations are all set the same:
/Volumes/Band-in-a-Box Audiophile/Applications/Band-in-a-Box


Also, it's possible that the StylePicker window is simply being hidden behind the GarageBand window for some reason. Have you tried just minimizing the GB window and all other windows to see if it's there?

I tested on macOS 12.2 with GarageBand 10.4.5, DAW AU plugin 2.12.15, which is what you have I think.


Andrew
PG Music Inc.