According to Apple's Wi-Fi base stations: Setting up and configuring a roaming network (802.11 a/b/g/n), there's an implicit assumption that I want to use a wireless for everything. That is, the extended base station backhauls its traffic using Wifi.
Wifi is a limited resource, and there's only so much bandwidth in a channel. I'd like to reserve the wireless bandwidth for clients, and backhaul the extended base station traffic using ethernet. But Apple's article doe not discuss how to bridge the APs using ethernet through a switch.
How do I set up roaming between two airports and backhaul over ethernet using a switch (and not Wifi)?
Related (from James' answer below): in the upper diagram, the extended Airport is plugged into the primary Airport base station directly. That's not quite the configuration we have because we can't run the direct cable. We are in a historic building, so we can only use existing cables and run it to the basement and into a switch.
Best Answer
The diagram at the top of the article you linked to in the question seems to show exactly what you are trying to achieve; the extended base stations are connected via Ethernet, from their WAN ports, to a network port on the primary base station. Have you tried this configuration?