post to a private feed?

First, see this article:
http://www.wpbeginner.com/wp-tutorials/9-extremely-useful-rss-tricks-and-snippets-for-wordpress/

For some more code:

You can exclude a category from the main feed by putting this code in the theme’s functions.php file

You must know the category ID of the category you are excluding. In the code below -5 means don’t include any posts in category 5. If there is more than one category they can be entered as
$query->set('cat','-5,-7');

add_filter('pre_get_posts','my_feed_excluder');
function my_feed_excluder($query) {
 if ($query->is_feed) {
   $query->set('cat','-5');
 }
return $query;
}

If you create a feed URL for one category, you might want to burn it at Feedburner. The reason is you can give it a generic name that does not include the WordPress category URL in the feed URL