Maybe try a left handed drill bit? We have them here at our truck shop and use them on occasion when a blot head snaps off. Usually will start a bit and grab the bolt and back it out.
Just out of curiosity which bow is it? If it's one with te plastic module then just drill the head off the screw. If you mess up the module I'll send you a new one.
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I second the left hand drill method. used it countless times in the power plant to even remove tough stainless steel fasteners that are extremely tough to get out.
If the screw is sticking out far enough out the back, try using plyers and unscrew it that way. If you can get the head far enough above the hole, then you could use the plyers on the head to keep unscrewing it.
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For those that can't picture it, this is what I'm working with. I'd like to get the allen head out and save the mod screw if I can - otherwise, I'll use the bit to drill out the screw itself from the backside and find a replacement.
We're you trying to tighten or loosen the mod screw when it broke? The left handed drill bit might break the head loose in the screw or at least back the screw out where you could work on it easier. Get it away from your string and cables anyway. Would be the way I would go at it.