I think I figured out what keeps happening with my sort values. Because with artists I use the literal artist value (Hank Williams instead of Williams, Hank), I don't tag the sort artist or sort album artist with Picard. Since I DO want composers sorted by last name, I was letting Picard tag the sort composer, though.
As a result, a file gets read in MusicBee with an actual Sort Composer in the tags but only an inferred Sort Artist/Sort Album Artist, and because the Sort Composer is an actual tag in the file it ends up being prioritized. That fits with what you said here, Steven: "directly reading the custom sort tag from the music file and adding to the sort lookup list in your musicbee but only if there is no sort value already."