Thanks for sticking with me, Jim. You inspired me to keep trying! So I have figured it out.
You would think that if you bought this combination from someone, they would have been using it. It is possible that the guy who had it before me could never get it to work and just sold it on, but he never said. It's been hair-pulling, but the good part is that I now know my equipment a lot better than I would have, if it had worked as soon as I hooked it all up.
It was a firmware problem. Turns out the FDM-DUO was several versions behind the current one, and probably didn't know how to talk to the ART. I had to go through all the upgrades, including putting the latest version of Flashmatic on my computer, which immediately killed all my PC's USB ports, and wouldn't have worked anyway. After removing all those bad drivers and installing an old version of Flashmatic, the upgrade went smoothly. When I hooked everything up again, it the DUO-ART recognized the #31 option from the radio, and it is all working just fine now!
This is such a sweet combination. I had an FDM DUO several years ago, but sold it, and bought a KX3. I just couldn't resist when a good deal came up for both the FDM and the ART. I think we'll all be happy together.