Would you be able to split the Sync Settings so that they're dependant on which library you're syncing from? Or even just Media Storage and Playlist Storage (if that's less work).
That way I'd be able to force-in the extra \Music\ folder where needed.
For what you're trying to achieve, I think that wouldn't be necessary.
If you change the storage path setting to use naming template, you will get more control over the situation than with "preserve folders and filenames".
So try switching to that setting and then use the following template for the "music files" section:
$RxReplace(<URL>,".*?(Music.*)","$1")
Edit:
Sorry, forget about that. It's not going to work. While the path does display fine in the template editor's preview list, the same cannot be said of the target filename in the sync preview window - MusicBee seems to replace the path's backslashes with a space.
MusicBee seems to replace the path's backslashes with a space.
I just remembered that in MusicBee, we can set the desired replacement values for illegal filename characters in Tools > Organise Files > Filename Char Mapping.
By default, the backslash is set to be replaced with an empty space.
But for the purpose of what you're trying to achieve here, you'll need the backslash (\) replaced with something like an equal sign (=) instead.
You only have to do this once. Then going forward, the below template will now produce a target filename beginning with the "Music" parent folder.
If you don't mind the workaround, I think this should work just like "preserve folders and filenames" would have worked.
$Replace($RxReplace(<Path>,".*?(Music.*)","$1"),=,\)<Filename>