Redeclare a plugins function/class in my theme?

You can extend the existing class to override a particular class method (function):

class Mod_Groups_Post_Access extends Groups_Post_Access {
    public static function wp_get_nav_menu_items( $items = null, $menu = null, $args = null ) {
    // ... your code ...
   }
}

Then you would need to call $instance = new Mod_Groups_Post_Access(); instead of $instance = new Groups_Post_Access(); so you can use the modified function/method via:

$instance->wp_get_nav_menu_items();

You can read up further on object inheritance here:
https://www.php.net/manual/en/language.oop5.inheritance.php