I investigated my previous steps and redid it all. The results below. For now for the lazy, the process works and works well - I am really grateful for the plugin! Doing everything like in the readme works great!
However, I did not stop there ;)
Deactivating the old U BuddyPress Forum Attachment plugin indeed as you suggested works.
However, deinstalling the old U BuddyPress Forum Attachment plugin leads to file not found errors. Maybe the readme could advise against this as for a non-programmer like me, removing and deactivating a plugin were nearly identical ;)
Reinstalling the old U BuddyPress Forum Attachment plugin afterwards makes it even worse. After clicking on Download the page reloads without an error but no file is served :/
Just for completion - deactivating your migration plugin is also not a good idea, since then the attachments vanish again ;) Maybe it could be mentioned in the readme that this needs to be permanently active?
In any case, wonderful work - really, really appreciated and works really well <3