Permission Denied (403) using /batch
I would look at your .htaccess rewrite rules. I've hit instances in the past where batch requests wouldn't fire during updates. Turns out I had custom .htaccess rewrite rules interfering with those requests. 

Based on your example URL, it looks like you're running Drupal in a subdir on the domain. Have you set up the RewriteBase directive for that subdir? Do you have any other custom redirects in .htaccess?