Solved together with support:
Create a DB-Field "Document Name" and check the button in the last column (file name).
You can fill this field with your desired fiel name via DWControl.
Then the value you've set will be shown and even used when the document is downloaded.
Caution: The target basket has to be bound to the right file cabinet that has the DB-field (here: DOCUMENT_NAME).
With DWControl always use the field names from the database (here: DOCUMENT_NAME) not the names that are
shown to the user at the UI (here: Document Name).