Turn off (suppress) Msg 3013 "BACKUP DATABASE is terminating abnormally"
There is no way to "suppress" the error, nor would I personally want to. The error is letting you know something happened and it's up to you to go figure out what that something was.

In your example, it says the database doesn't exist. Let's say you lost connectivity to disks, or there was corruption, or ... ad nausea. In this way, you can key off of the 3013 error as a general "backup database issue" (alert) and then investigate further.

Assuming you could suppress the error message, you'd either have to have an alert or line item for each individual error that could possibly happen or just go about your day until someone or something complains. Either way, not so good.