Answering my own question... It's risky using JB Weld or something similar, because it'd be really easy to bond the remains of the pilot screw to the carb housing. I wound up drilling a 1/8" hole down the axis of the screw, and using a fluted screw extractor. However, this type extractor is designed to be driven into the pilot hole; if you use it to pull on the screw, it tends to unseat itself. This turned out to be a problem, because once I backed out the screw past its last thread, it was still held snugly in place by its o-ring. I was eventually able to use some fine needle-nose pliers, with one tip inside the drill hole, to grasp the pilot screw and pop it out. I'm not sure if a spiral screw extractor would've worked better; it might've just spun the screw loose, rather than gripped well enough to have allowed me to pop out the pilot screw.
Regardless - the degree to which this procedure is documented is absolutely pitiful. I can see Honda not wanting you to mess with the adjustment of this screw (CARB reg's?), but I'd expect the Honda shop manual to give more guidance on how to remove the screw than just "...remove the broken pilot screw". I won't be replacing the limiter cap, in case I ever need to clean the carb again.
NOT SO FAST... while re-installing the pilot screw, I discovered the threads in the carb body are buggered up. They're extremely fine, and I probably damaged them while driving the screw extractor into the pilot screw. Looks like I'll need a new carb. What a bummer!