Drupal 7 module hook_preprocess_page &$vars
You are trying to render $page['branded_footer'], but the module preprocess function is setting $vars['branded_footer'], which would be accessible as $branded_footer in the template file.

$page['branded_footer'] would be set if the theme defines a branded_footer region. In that case, you should follow what How do I create custom regions in a theme? says.