.bbl is the backup baseline file, which contains all initially backed up tags. all other backups are incremental (and these files have extension .xml), i.e. contain only tags, changed since the baseline backup. you won't see this file in plugin dialogs (1st visible backup is empty incremental backup), but you can easily find this file (and all other, incremental files) in backup folder.
but you won't be able to restore tags on mb, which uses a new library. to restore tags, you must copy .mbl file to the new pc, but this will automatically keep your tags.
the only solution i see is to copy mb library yo the new pc, and to use file> library> relink music file paths.
the "backup & restore" part of the plugin is intended to be some kind of "time machine", to use it always on the same mb library (just because the plugin needs to know tracks unique ids to restore tags, using track paths would be too unreliable).