Somewhat good news! I connected a jumper wire from one of the red "hot" wires to the voltage regulator where the brown wire connects. The charging system now works! I then hooked up my rally gauges again, and sure enough, it is showing a charging state.
BUT (there's always a but), I still haven't figured out why that brown wire isn't getting any power to it. My next step is to pull the gauges out to get to the ignition and see what's going on back there. Maybe it's been disconnected for some reason - who knows.
If I can't figure out the brown wire, is it ok to keep that jumper wire hooked up to the regulator, where it will always be hot? Or will this somehow cause a problem?