Well this is finally resolved. After another nasty letter which said they'd send round the debt collectors plus a few emails later, I finally got them to understand that I hadn't received an invoice. They then emailed a copy of the invoice to me and I promptly phoned the magic number and paid the damn thing.
Why couldn't they have done that in reply to my first email?
Very poor customer service, but then again I'm not actually a customer so I dunno.

The result is next time I order from Chai I'll ask him to send via a different courier.