In order to save  list as template, you must be a member of a site  group with the Manage Lists right on the source site and on the destination site. By default, the Manage Lists right is included in the Web Designer and Administrator site groups. If you don’t want the user to access  the top level  site, you need to have elevated access to the List template  gallery, break the parent permission for the List template gallery, and add the user as Designer to the library.