Well that's still using php-cgi and it's still too slow!
But I have solved it in a different way: by using ACL.
The webroots of these sites are set to user:group www-data:www-data with an ACL entry u:web29:rwX,g:client5:rX and it includes a default.
What I did was, I created a script to do all of this for me hourly through a cronjob. Here it is, I hope it will benefit others with the same problem:
# Fix permissions for webroots that are using mod-php and wordpress.
chown -R www-data:www-data /var/www/site.org/web
setfacl -bR /var/www/site.org/web
setfacl -Rm u:web29:rwX,g:client5:rX,d:u:web29:rwX,d:g:client5:rX /var/www/site.org/web
# Repeat this for all the sites that need it. Crontab it for your own convenience.
# New files will automagically inherit these permissions.