ISPConfig 3, a ninja sets my open_basedir
Was much easier and obvious than expected.
In the "Options" panel of the "Web Domain" in ISPConfig there is a textbox labeled PHP open_basedir. I still don't know which exact config file that is going to create or edit, but it works.
Also worth mentioning for this textbox - the default value when creating a new site can be changed in:
System >> Server Config >> Web >> PHP Settings >> PHP open_basedir
Thank you so much for the answer! I looked everywhere, and the answer was right there on the options screen! Thank you, thank you, thank you!
After further investigation, ISPConfig executes php-cgi when a vhost starts. The starter script is in /var/www/php-fcgi-scripts/web?/.php-fcgi-starter, where the ? represents the site number within the client. The open_basedir is passed as a parameter to the startup. That's why overriding in the individual php.ini doesn't work. At least, that's how it's configured when the site is fast-cgi / su-exec.