I used Keyboard Maestro, which (among many other things) lets you link the script to a hotkey (like the default ⌘D for done tasks).
There should also be a way to save it as a Service, which you could then assign a hotkey in System Preferences. I’m not sure how to do it myself, but I believe I’ve seen someone explain it in one of the other script posts in this form. (List of the script posts is here)
Zip of the script for download:
Add Timestamp to Done Tasks.zip (1.2 KB)
Hope this helps!