Is there a hotkey for moving items between your stash and inventory for controller? As far as I can tell, there should be plenty of available buttons for this functionality. Is it possible to add this functionality within the game somehow or with the help of user-made mods?
Not at the moment. The fact that there even is a controller option is only due to one dev messing about in his spare time and was never intended as an option in the beginning. The good news is that due to the console port the controller is going to have to be changed for that and we shall see what happens with the PC.
If you are willing to use user-mods (there are none that do this) then you are willing to use a program like xpadder to bind Shift-click to whichever button you want. http://xpadder.com/
Handy lil program.