Ah. Figured it out.

It's all about the MAC ADDRESS CLONING.

See, I'd set up my Actiontec router to only accept certain MAC addresses. It all worked, until I fed it this thing, which (by default) clones the wired MAC address into the wireless MAC address. And I hadn't put the wired MAC address into the acceptables list.

So it would connect to the LAN with its initial wireless mac address, but then get no further because its cloned address wasn't allowed access.

So now I've disabled the MAC address cloning on the thing, and it seems to work, but sometimes its settings don't stick. Not sure how to trust that. Hm.

Special thanks to Drakino, who, in a two-minute IM conversation, let me bounce my complaints off of him and suggested that I try the thing without security to see if it worked.