Thanks, we got a good backup without the error last night, so I'm hoping we've fixed it permanently. I ran "vssadmin list writers" on the mail server after posting that, as suggested above, and could see Microsoft Exchange Writer reporting a failed state. We removed a couple of databases we had previously dismounted because of errors, and then restarted services as suggested above (veeam-backup-replication-f2/upgrade-to-v9-and-exchange-2013-dag-backup-warning-t32764-15.html#p181174
), and then Microsoft Exchange Writer was no longer reporting a failed state. Then the evening backup was ok.
I'm not sure which of those actions fixed it. It might be that I forgot to restart the services after dismounting the databases, and that removing the databases wasn't necessary.