Ok, I have resolved the issue, after consulting twice with D-Link support (who were ultimately able to successfully identify the problem).
I had to change the switch that the system was operating through. The original switch I was using was a D-Link GO-SW-8G 8-port unmanaged gigabit switch. When advised of that, D-Link Support responded with the following:
"Unfortunately, we have reported cases that Covr backhaul will not work with GO-SW-8G or any older D-Link switches that uses [sic] IEEE 1905.1 control packets with 01-80-C2-00-00-13. DGS-105 and DGS-108 and [sic] work fine with the Ethernet backhaul."
After receiving that news, I changed the switch to a Netgear GS308v2 (which I happened to have on-hand), and the problem was immediately resolved. The nodes switched to solid white LED indications, and the web interface now reports an "excellent" connection.
I'm not sure how to determine if a switch uses the protocol/control packet scheme they describe, but it must be listed in the specs somewhere. My Netgear just happened to work.