Symptoms
Additional PHP version is installed according to KB #118378 and registered as FastCGI handler in Plesk with no issues.
However it is not available to select in Plesk in Home > Subscriptions > domain.tld > Websites & Domains > Hosting Settings in PHP support
Cause
Apache mod_fcgid module is not installed
Resolution
Install Apache mod_fcgid module using Plesk autoinstaller.