I don't understand what the Site Export extension's author wants to say where he's talking about the cronjob in the readme, and I can't find the mentioned "cron.php" file.
Anyway, thanks for the suggestion to look into the ajax.php file, because I'm thinking that I could start commenting out the part that deletes the temporary .zip file that is created on the server when an export is completed, and go from there creating a script to automatically get it and do the subsequent operations.
2 beers!