EDIT
Clarified question: is it possible to distribute backup copies of applications on Google servers.
My answer: AFAIK is not possible because Google will keep backups for each package name, which must be unique for each application.
Original answer
I'm still not sure why you want to access the same data from both applications. Yes, I understand the update path, but why not just copy the data when the user receives free advertising?
In any case, your BackupProvider implementation should be responsible for managing shared data. Now, since you will have 2 versions of the provider (free + paid), they need to make sure that they do not overwrite each other in a common location (for example, sdcard).
But even if you follow this approach, Google will still support two separate copies of your user data, one per application (AFAIK this is determined by the name of the application package, which must be unique)
source share