When your client first opens a workbook and connects to the tde file, have them use 'Extract' instead of 'Live' option. (This will extract the extract.) Have them save the workbook as a packaged (.twbx) workbook. This will collect up files (including the new extract), packaging them into what is essentially a zip file. This should free up the original extract to be overwritten.
When the client wants the latest data, then all they need to do is 'Remove', then 'Remove and Delete' the extract (which is the copy) and Tableau will automatically reconnect to the original extract. Of course if the client forgets to re-extract the data source, then you'll have the same problem. It's easy, only taking a few clicks, but it may be hard to get them to do it.
I'm able to create the packaged workbook as described in your first paragraph. When I saved the tde file as a packaged workbook, it asked if I wanted to create an extract which I did. The other option was to cancel and connect live to my data source. When I right click on the extract table in my main window, I have the option to duplicate or remove. Is this the 'Remove' you suggest I perform?
I still show connected to my original tde and my Extract (Extract.Extract) still shows as a table but I do not see an option to 'Remove and Delete'.