When I click on the button after Preferences > "Automatically check for updated translators and styles" I get an ERROR response (this happened in v4 and v5)
This works in Zotero generally, so this is likely either a conflict with another extension (I'm not sure if BBT overrides this) or a problem with your network connection. If you can't figure it out, I can look at a Debug ID for an update attempt.
The Debug ID is D1839283797.
Thanks for looking into this.