I suggest...jam a screwdriver instead of a saw blade down into the cavity. Use several screw drivers if you can. Get it jammed in there tight. You might also try shoving a dime or penny or washer or some additional kind of spacer to hold it tight. Turn it to the right a tiny bit to see if that loosens it. Then left.
If you have to...drill a hole in the funny nut, and put a pin in it, or two. Grip pins with vise locks.