User mailbox full, archive full.

That's great.

I would personally reach out to Microsoft Support over the phone to request an update on the ticket and explain the urgency of resolving before mails get rejected. I think that an answer can be found in the article that Adam provided - https://blogs.technet.microsoft.com/exchange/2018/04/09/office-365-auto-expanding-archives-faq/ - the main archive could contain ghosted content that has been copied to an aux archive but has been kept for 30 days before it is flushed. However, that's difficult to tell here.

I would be very interested to hear what the root cause of it was in the end.

Best, Chris