You might start with small amounts of penetrating oil carefully applied, then take a dremel and (very ) carefully grind just enough of a flat into each side of the stud so you can get a channel lock on it with enough pressure to grip but not so much that you'd bend the stud. That might give you enough purchase to unscrew. The wood might take a few small dings depending on how but if its already been worked on, just smooth with sandpaper and seal. You could also take a dremel cutting wheel and cut some shallow notches, so you could use a screwdriver on it.
Edit: looking at it from the top down view, it does look like you have enough clearance between the top of the stud and the tip of the screw to use a dremel cutting wheel to make a notch on each side of the inside lip of the stud without cutting into the screw itself. Cut them to the appropriate depth and width for a screwdriver and then at least you have something to get a grip with without worrying about marring the wood as much.