[el-Odysseas]

Note that I do allow multiple tempos to be added to the metronome, so at the very least, if a new tempo is added I have to save it in the song. ...Obviously the simplest thing is just not to save the tempo changes at all, but then you wind up with that weird problem when adding additional tempos in the metronome dialog.

I didn't know, that you can allow multiple tempi to be added to the metronome (with a dropdown-list of all the inserted tempi)! This is nice!

The reason adding multiple tempi to a piece/a song, is that the piece requires it: the beginning should for example be played Allegro, the middle part Adagio etc.  
These tempi are still a part of the song, so I do not find a reason to constantly change them and therefore no reason to automaticaly save the tempo-changes in the metronome. You enter the song-tempo (or multiple song-tempi) in the tempo field of the edit menu and these do not change, unless you correct them, or update them.

In this case a button to synchronize/save changes on the metronome interface would be great. When a piece consists of multiple tempi, you should be able to choose the tempo  that you want to edit (from the dropdown-list showing the different tempi), tap or insert the new BPM-number and press "synchronize/save changes".