May need to damage the wood a bit

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but you could cut around the screw a bit to allow a needle nose vice grip to dig in and clamp down on the screw head then twist it out.

You could also get a hax saw and saw a slit into the head big enough for a reg head screw driver to get in there and screw it out.

All would require repairing the hole when the screw is out, but you would have to anyway.