I would cut off the head of the screw so you can at least get the pickup out. Then try removing the threaded portion that's left with pliers. At least without the pickup in there you have a bit more room to manoeuvre.

Worst case scenario I can see, is having to pull the threaded portion out by force, then carefully drilling and plugging the bottom of the cavity for a new screw hole. There may even be enough room in the cavity that you could get away with just gluing a thin wooden block over the old hole and fixing the adjustment screw to that.

I can think of more elaborate possible fixes using threaded inserts and swapping machine screws for the timber screws, but I can tell if that's feasible without seeing it first hand.