OP here. The short story is that it doesn’t work, at least not yet. The caveat is that it could well be because I’m not smart enough.
The device that my ISP calls a modem has two ports that I will call ethernet ports. One is connected to an Airport Express, and that is how I get internet. I just connected the other port to an ethernet to USB adapter and plugged it into a USB hub. I got a notification that there was a new device and I should configure it in the Network control panel.
In the Network control panel, I clicked the + at the bottom of my networks and selected “USB 10/100/1000 LAN (en6)” from the dropdown menu. (I was also offered “USB 10/100/1000 LAN (en3)” from the menu; “USB 10/100/1000 LAN” is the name of my other connection.) I left untouched the “Using DHCP” choice for Configure IPv4. After a while, the indicator changed from red to orange. Under Status: Connected, it said that the connection had a self-assigned IP address.
I switched the ethernet cables (ISP device and Time Capsule). After a short while, the new network had a green indicator and the old network had an orange indicator and the message about a self-assigned IP address.
I told Time Machine to back up to the Time Capsule and the backup started. As I was typing this, I got a message that the backup failed. I switched the cables back and told Time Machine to back up to the Time Capsule. The backup started and shortly thereafter I got a message that Time Machine completed a verification of my backups and (to improve reliability) it must create a new backup for me. I seem to get this message about every six to eight months, and it seems like it was due, so I don’t know if this is related to the “two ethernet networks” scenario or not.
I disconnected the Time Capsule ethernet cable from the adapter and told the new network to Renew DHCP Lease (in the Advanced… pane). After a short wait, it had a self-assigned IP address. I reconnected the Time Capsule and made the new network inactive.
I would fiddle with it but it seems more productive to ask here for suggestions, especially for configuring the new network. Thanks for any insight.