In this thread the poster was getting 403 Forbidden when they tried to save admin settings. They contact the host who said "There was a ModSecurity rule that was being triggered and causing the 403 error. I have now whitelisted this for you." – this fixed the problem. Maybe there's a similar issue with your host.
When I suggested changing $CFG->admin to 'admin_moodle' maybe the host is blocking "admin"? You could try again but use $CFG->admin = 'manage' and rename the admin folder to manage.