If you untick "override default folder" for a podcast, instead of immediately applying the settings in Library Preferences, MusicBee first uses the default podcast organization (<Publisher>\). You have to go into preferences and use the Reorganize button before it will apply any custom organization template. And it still shows ...Podcasts\<Publisher> as the default path in the podcast settings, despite any changes in library preferences.
Also, reorganizing a podcast by unticking "override default folder" causes playback (not of the podcast) to stop. Very strange.