You could use VBA to do this, there is nothing that I know stored in the document.
You would save a document variable and change it upon closing the document.
This is beyond the time I have available and may be beyond my expertise, but that is how I would approach it.
The editing time set by Microsoft is pretty meaningless in ordinary use.
The following is a simple macro I use for a basic time stamp in documents. I attach it to a QAT button and in some templates to macrobutton fields.
Code:
Sub DateTimeStamp()
' Inserts basic date and time stamp as text
' HansV Eileen's Lounge
'
Selection.TypeText Format(Now, "dddd, d mmmm YYYY - h:mm AM/PM")
End Sub