FWIW when faced with this situation I created a new admin role "Mover" which has only one privilege: the ability to move files and folders to shared drives. I assign the role to whoever needs it, usually with a specified time limit. I would just go right to the admin console, create the role and assign as necessary. Doing this with GAM is probably possible:
https://github.com/taers232c/GAMADV-XTD3/wiki/Administrators but I'm not sure and you would really have to get into the weeds with the BNF and API doc's to try and sus it out. There is no admin level privilege, that I'm aware of, that restricts a user from deleting files, those privileges are managed based on membership on a particular drive.
That's my 2 cents worth. Your mileage may vary.
-Marsh
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Maj Marshall E Giguere
NH Wing Director of IT
Nashua Composite Squadron IT Officer
Civil Air Patrol, U.S. Air Force Auxiliary
Volunteers serving America's communities, saving lives, and shaping futures.