As I've gotten more involved with this, some additional details have come to light that make it...complicated.
First, to answer these questions: there is no DHCP running on the RUTX10. The devices have been assigned static IP addresses that are manually configured. These are not static leases assigned via DHCP. They are on the same range as the subnet that the RUTX10 is on, i.e., the router itself has an IP address on that range, as do the devices connected to it. The RUTX10 is just a bridge connecting some wired devices to a wifi network.
I've thought about an IP conflict as well. Here's what makes me think it's not that: I can reach the devices via their web interfaces with no problem. It's just pings that are not working. I would expect that if there was a conflict, I would not be able to reach them. But I also know that IP conflicts are unpredictable, and we hunted for this problem to rule it out.
Here's the thing that makes this more complicated: there are two devices connected to LAN ports on the RUTX10, but only one of them is having trouble, i.e., the other answers pings (sorry, this is not what I posted in the original question—I misunderstood what was happening). I would think this was a device issue, perhaps with firmware or something, but the fact that I can plug into a LAN port and ping it with no problems from another device makes me discount that possibility.
I have four of these setups out there, and they are all behaving this way. I will be doing some additional troubleshooting, for example placing a switch between the devices and the RUTX10.
Thanks for your reply!