The only thing that I can think of is that the ATF in the transfer case might've thickened up. It's happend before, and when I lived up the street from you (eileen) I had heater pads on EVERYTHING.
I suppose that if the fluid were in a syrupy state (I've seen 10w-30 freeze into Jello), it could prevent the electronic clutch from engaging.

-50 does some pretty weird stuff.

Even an auto transmission car will become sluggish at these temps, but the fluid cannot physically impede the meshing of the gears.
In the case of our TOD case however, the mush could prevent the clutch packs from touching and transferring torque.
You might think that it would warm up due to friction, but with temps like that, nothing *really* warms up at all.

I say put a small (80-100W) silicone pad heater on the transfer case.