You can drag to a menu; you can drag to the dock - but can you drag to a Dock Menu? eg to open a document different ways based on which item in a dock menu you drag to.
I don’t believe so, no.
Since when can you drag to a menu?
Look at this:
It uses Accessibility API to locate the position of its icon on the Dock, and then draw a floating window which looks like dock menu. By the way, how can it use accessibility API when I disabled “access for assistive devices” in System Preferences?
Use AXMakeProcessTrusted or
sudo chown :accessibility myApp sudo chmod g+s myApp
If you must, then authenticate and use a small helper program to do the dirty stuff, but don’t make the gui run as superuser..