To change or extend the functionality of WordPress Core, WordPress relies heavily on actions and filters. I’ll assume you have basic knowledge of this, and if you don’t, you will after reading up on it in the Codex.
render_screen_meta is a public method of the WP_Screen object. You can’t overwrite is by defining the function somewhere else. public determines the scope of a method or property in class-context. If you would like to read up on this, and on classes and objects, I suggest checking out the PHP manual on them.
There are many filters available to modify behaviour of WordPress Core behaviour, including a filter to change the Screen Options contents. This filter is called screen_settings, and it is located in wp-admin/includes/screen.php:990.
Basically, it allows you to completely modify the contents of the Screen Options. Using it, you can add the switch yourself:
add_filter( 'screen_settings', 'wpse148978_screen_settings', 10, 2 );
function wpse148978_screen_settings( $screen_settings, $screen_instance ) {
$screen_settings .= 'Wow, a button! <a href="#" class="button button-primary">Switch!</a>';
return $screen_settings;
}