Ƭᴇcʜιᴇ007 is unfortunately correct: there's no way to set up a rule that will permanently delete an email while Outlook is not running. Your options are either a) create a client-side rule in Outlook that will process all the permanent deletions at once when you next open it, on your return (prior to which, your inbox will fill up with crud), or b) create a server-side rule, via Outlook Web Access, that will delete those emails so that, when you next open Outlook, they will be in your 'Deleted Items' folder. The latter is likely preferable here.
Note: I realize you said "there is absolutely no opportunity to use a different client at all ever", but it's unclear whether that includes the web-based version of Outlook Web Access (since, after all, that's still Outlook). OWA comes with Microsoft Exchange and while I'm sure it's possible that there are organizations out there that use Exchange but do not implement OWA, I've yet to come across one.