So as we'd expect, a cron job does the trick. Perhaps not the most elegant, but it works. The cron job copies index.html to index.php. Then in .htaccess, just to be safe, I put this:
DirectoryIndex index.php index.html
simply to be extra-sure that the php file would be used in preference to the html file.
I used the -u parameter for cp (ie: cp -u index.html index.php) just to prevent unnecessary overwrites of the php file.