I don't understand how permissions are working. Many times I have to setup 775 permissions to run many scripts. In Joomla, after a component installation, its folder owner is www-data and not client7.
There are no directory permission cahnges required, you used just the wrong website settings. Joomla and other cms system require the following website settings:
PHP mode: php-fcgi
as described in the ispconfig manual.
The chown / chmod that you did is insecure and can cause that other sites get hacked on your server. You shoudl undo these changes and chnage the settings for the website as I suggested above.