I made a small update to my code to get around an error caused by no model name being displayed when reference display is turned off. For anyone that stumbles on this and wants to use it.
Sub SetLogNameAsDefLogName()
Dim att As Attachment
For Each att In ActiveModelReference.Attachments
If att.AttachName <> "303881-Water-Profile.dgn" Then
On Error Resume Next
att.LogicalName = att.DefaultLogicalName
att.Rewrite
End If
Next
End Sub