Success!
I knew what I had to do: I updated the firmware on the A unit from 1.02 to 1.04, then used Ethernet to reset it to factory settings. Then this morning (when the whole family was offline), I unplugged the old Apple router and started the installation process from scratch, carefully following the instructions to a T. Voila!
The second unit two floors away is also up and running with very fast throughput according to the speed test.
Terrific!
Thanks for your help!
One more random question, which I can't find the answer to anywhere else. Is there a difference between the A unit and the second unit? Or are they interchangeable?