MusicBee has recently added native support for TOPE (Original Artist) and TOAL (Original Album). I see in the documentation that Picard supposedly writes Work to TOAL, although I can't tell that that's happening.
In any case, the problem is not with what Picard is writing but what it is reading. The TOPE and TOAL values that MusicBee writes are not showing up in the bottom panel. I checked with Tag & Rename to confirm that they were actually there, and they are. There's nothing in any of my scripts that would affect these values, nor do I think there's anything in the plugins (though I could be wrong about that). The values ARE being cleared by Picard, according to the "clear existing tags" setting.