[ipadonice]

 I just said if you don't want to use the built-in synchronization functionality, you will just have to sync the database file between devices (at a bare minimum). 

My fault, I didn't get that point. 
I'm experimenting with a test database and I'm trying to implement your recommendation, but I'm running into problems:

You just need to have all your files under one folder, and you change the storage location to that folder. 

This works on my Android device. 

If you then import those files into MobileSheets, it will recognize that they are already under the storage location and it won't move or copy them. So then you get the best of both worlds - you can manage your own files under that folder with no files being duplicated, but also easily synchronize the database between devices. On iOS, due to the restrictive file system, you already have to keep everything under the MobileSheets folder, so you would just have to ensure that you have the same folder structure under that folder as you have on Android.

I'm not sure, if I'm doing things right here: 
Database and sheetmusic are in the same folder due to the custom database-folder, that I selected on  my android device. I synced that folder to my iOS device. But: On iOS I can't select a custom database-folder, because this configuration is missing. 

Edit: Seems like I‘ve hit the borders of iOS quite fast, this is my second day with the new iPad. I tried to start a new library.
It‘s not possible to not make copies because of the right-system I think. In MS is no option to select if the app should manage the files or not. Due to this it‘s not possible to select a whole folder for importing if it‘s not in the „on my iPad“ selection, even if I select another location (the same where my sheet music is) for the app in the configuration. I also tried importing via dropbox, but it did only import 5 files out of 21 without more information. Is there maybe a logfile for it? 
I also tried to sync directly into the MS folder, but the sync app (Synology Drive) makes it impossible to select a local folder. It always syncs into an own space on the iPad. 

This is really frustrating - I‘m not mad at you at all, it‘s probably the iOS restrictions making things difficult. 

I think I will stay with one device (iPad) and will manually copy files from the synced folder into the „on my iPad“ folder  as long there is no better solution for this. :-(