Clean way of Extending Stud for Mounting on Drywall
I know that your desire is to not mount any more wood but from the description that you gave it seems like the very best choice is to actually mount a 1x4 or 1x6 board across the width of the wall and screw into the various studs. Such board could even be full room width or at least longer than the backdrop width. 

If this board had the screw heads counter sunk and then filled in and then sanded, primed and painted to a color to match the wall it would be better looking that you might imagine. 

This board would give you ultra flexibility where to mount the hooks that the backdrop hangs from.