So far, so good. I was easily able to find a replacement C13/C14 receptacle, 14 gauge wire, female spades, and various diameters of shrink wrap. At this point, I've soldered a bit (about 1") of extra 14 gauge wire to the 12 gauge wire coming from the bike (to make up for the lost length on the old spades). I've put the three resulting wires and spades in some small shrink wrap and heated them closed. I've attached the spades to the new socket, and I've plugged the unit in, and switched on the power. After about 5 minutes, with my fingers directly on the shrink wrap, there is no detectable warmth coming out of any of the wire/spade/wrap units.
I still have to shrink on some 1" diameter wrap to weather-proof the whole bundle, but I'm wondering if I should stuff anything in there first? The old unit seemed to have some sort of solidified gunk in between the small wraps and the big wrap.