Restore a database backup on another server without sysadmin account
I think your main issue is that after restoring the database to your new target user, that your account "WebUser" is effectively orphaned.  

You should have your DBA change the SID for WebUser (and any other user that matters) to be the same as in your source server.  

Note that this will require dropping the user and re-creating it in the target server.  And any other databases will need to be fixed as well.  

But this will allow you to restore the database from the source server, to the target server and not have to worry about permissions when you get done.