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Centerpunch the screw head, then drill the head off with a drill bit the diameter of the screw shank. Pull, pry, wiggle, the board free, then put visegrips on the shank protruding from the bottom piece and unscrew itAlso, there's s company I can't remember offhand (see "CaitD1below; she says Woodcraft sells them) that makes what is basically very small holesaws out of hardened steel tooling, which I found online. They work, but they're very brittle and you need to be careful how hard you tighten your chuck. For a single use you can ma,ke one from steel tubing and a file..
...and another thing; though i've only seen them on bronze wood screws in wood boats, there's a cross-headed screw called a "frearson" or "Reed & Prince" which looks VERY much like a phillips screw. The tip angle is different, though, and the four 'lands' of the tip don't taper like Phillips do. There are 3 sizes of frearson drivers, but they all can be used in any size frearson screw (or so they say...) Don't waste time looking for them in hardware stores; I just went through that.