No, sadly, no resolution. I use stock Play Music on my phone.
What I gathered from other threads is that Android doesn't track play counts, so there's no way for MusicBee to tell if a track has been played. With iTunes, I was using iSyncr, and that did the trick, so it must be using it's own internal trick to keep track of plays. There is no equivalent app for MusicBee to tell the desktop that a song was played, and it's not stored in the tags.
Since I didn't get any response from Steven (presumably, because it's out of scope with what Musicbee supports), I've learned to live with it. What I gained from switching to MusicBee outweighs this one negative, and I've learned to just consistently bring my iPod Classic with me to the car (I listen with the desktop client at home). I still listen on my phone sometimes, it's just that my dynamic playlists that use play count as a criteria aren't going to get updated. Instead, I just sync over playlists with my highest-rated songs, or new stuff - playlists not affected by number of plays or date last played.