Moving the Sites/Default/Files Directory
Look in Admin > Config > Media > File System:

/admin/config/media/file-system


  Public file system path
  sites/default/files
  A local file system path where public files will be stored. This directory must exist and be writable by Drupal. This directory must be relative to the Drupal installation directory and be accessible over the web. This must be changed in settings.php


So there you have it. If you want to change the file system path for the public:// stream wrapper, you will need to edit settings.php.

EDIT: I'm not sure what else is at play here. If you still can't get this working to your client's specifications, you may be able to spoof it with a symbolic link in the file system:

ln -s files sites/default/files


The files would still be hosted in the wrong location, so this is actually no solution. Hopefully somebody else can point you in the right direction. Good luck!