I have altered the code using to just exclude the one file and that works. I'm not sure how to use a wildcard with this syntax so I still have to use the whole file name, but it's shorter code to accomplish what I need than I started with.
Sub SetLogNameAsDefLogName() Dim att As Attachment For Each att In ActiveModelReference.Attachments If att.AttachName <> "303881-Profile.dgn" Then att.LogicalName = att.DefaultLogicalName att.Rewrite End If Next End Sub