A quick follow-up. The site is now working.
Dreamhost have replied to our support messages to say they can't be sure the backup was current, or that the restore process was fully successful, and we should check the entire site for missing or outdated files.
All support messages throughout this process, either by email or on the Panel, seem to have been sent from some alternative universe where our website was fine, but experiencing brief periods of inaccessibility or slowness.
In real life, the site was completely and totally unavailable for 48 hours.
If DreamHost messages had reflected reality, this would have been annoying enough, but the difference between what they were telling customers and the truth made it incredibly frustrating.