I'm not really sure if it's a bug or not, so I made a gamble it's intended, so here's my suggestion:
Currently, when you remove a song from the songbook, it vanishes for good (At least, that's what happened to me, earlier today)
What I suggest is that what actually happens is that the song is "torn" from the Songbook, and shows as a song sheet in the PC's inventory.
You can write a song sheet.
You can remove a song entirely.
Remove the song in that order if that is what you want.