In reply to:

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(I think a Dremel cutting tool should work; these aren't exactly hardened steel screws we're taking about here.)

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Yeah, a dremel cutting disc will go through a mild steel screw no problem. Make sure you thread a nut onto the screw before you cut it - this serves 2 purposes. Firstly, it gives you some way of securing the screw whilst cutting, without damaging the thread, and more importantly, when you remove the nut, it will recut the thread at the very end of the screw where the cutting caused it to burr.