You're just going to break another screw off if you don't replace that one. Especially sailing in the ocean in Hawaii.
To remove the broken machine screw, use a small hole saw or plug cutter to remove the material around the screw so you can get some pliers on it to remove it. After it's out, you can fill the space with chopped glass / resin and reinstall the gudgeon before it cures. No drilling, and the resin will conform to the back of the gudgeon.
If you can't get it out with pliers, then you'll need to do some deck surgery - cut a small port into the aft deck to access the aluminum plate that the machine screws are threaded into. You might have to remove the plate to get the screw out, or at worst, reposition it and drill/tap all new holes.
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