I don't know what size machine or gun screw your Syracuse forend iron needs, but 3/16"- 32 falls between #10-32 and #8 screws. I believe the forend iron is what you are calling a "carrier".
When I have a missing gun screw, I've found I can carve a wooden kitchen match or similar softwood dowel to fit loosely in the threads with a slight taper. Then I wind a small strip of aluminum foil clockwise around the little tapered wooden dowel or match, and screw it into the threads. Several wraps of foil is needed. Then I can unscrew it and have a pretty decent impression of the threads, which I can then measure with a digital caliper and thread pitch gauges. Once you know the diameter and thread pitch, Google can be your friend to determine what size screw you need.
