I finally got this working, the main change being to /etc/apache2/mods-available/suphp.conf:
AddType application/x-httpd-php .php .php3 .php4 .php5 .phtml
AddType x-httpd-php .php .php3 .php4 .php5 .phtml
The only remaining problem is a constant stream in the apache log of:
[warn] Cannot get media type from 'x-httpd-php'
I might resort to installing ISPConfig on another machine somewhere just to be able to check the .conf files to figure out what got screwed up.