There should be no need to destroy the threaded rod to remove it. If you grab the exposed center section with miniature locking pliers, you should be able to turn it. Continuing to do so, about 1/4 turn at a time, should eventually expose one end far enough to grab it with the pliers (or a drill chuck) and screw it out completely. Peen or dress any protruding damage to the threads, make your wood repairs, drill through the old hole and new wood with a bit just large enough to allow the rod to pass, coat it with your favorite glue, and press into the hole. With the right glue, this will be stronger than the original and visually identical.

Brownell's sells 3/32" threaded pins, but the quantity (12), price, and shipping makes this uneconomical when only one pin is needed. STOCK REPAIR PIN KIT | Brownells