Very cool, sounds like it should work, if cooling the glass is such an issue, adding a cheap 40mm fan or 50mm blower to add some air flow or even a fan blade to the tool bit could keep the plastic from heating up to a critical level. Maybe also making lighter passes might help with pauses inbetween each pass to let not only the glass to cool but also the bit you are using. Now, I've never tried applying a light stream of water to plexiglass while cutting it but with other melty material it seems to help out quite a bit, just be wary that water damage can be an issue so throwing some saran wrap/ stretchy plastic over bearings and electronics before cutting could be a fairly extreme solution.
Thing is that there is a wide variety of plastic glasses, it could be just that this type of plastic or the type of bit you received were defective albeit a rare and doubtful case.