After my last post, I did more searching and discovered the 2.23 firmware upgrade. I applied that to my MiFi and ever since it has been unable to connect to the network.
The WiFi connection is fine. I can get into the admin page, but the device is in the disconnected state and the system log shows repeatedly "LTE Connection Failed." If I force it to EvDo, then that fails similarly.
I called up support and they said they could do nothing but swap it out. I even took it to the local store and they said the same thing and set up a replacement.
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EDIT:
It turns out that jeffr in another thread managed to save me. The MacOS X firmware updater is BROKEN, and the fix is to use the Windows one to re-do the procedure. That worked and got my MiFi back online.
The bummer now is that I'm still out $7, because it's apparently too late to cancel my replacement order. I'm not sure whether to accept the replacement or refuse it.