Like you mentioned in your manual process, you have to save out the family with a new name in order to “duplicate” at the Family level. You can likely still automate this, but you’d have to follow the same process:
- Isolate door family
- Save family to desktop with new name
- Load family from location (desktop)
- Delete file on desktop
These are just the first related topics I found but they should at least get you started.
SAVE FAMILIES IN FOLDER WITH DYNAMO v.2.1 - Revit - Dynamo
Load multiple families from a folder - Dynamo