With API you can get first the drawing (swApp.ActiveDoc) then drawing sheet (swdraw.GetCurrentSheet) and All views(swSheet.GetViews)
From each view (or maybe 1st view in your case) you can get a referenced model (Set refModel = swView.ReferencedDocument)
And then using CustomPropertyManager get the required property and rename the drawing using it.
I have it working. After I have it cleaned up I will post it.