You need to be careful using heavier oil in the t-case, its not splash lubricated like the old days, it has a gearotor oil pump on the output shaft with a fairly small diameter suction pipe and screen, the pipe is close to 6" long and I doubt more than 1/4" inside, imagine trying to suck cold motor oil through a straw, basically the top bearings would run dry until the oil warms up enough to flow through the pickup screen and tube. That said, I have been running 5w50 synthetic in my Dakota's t-case for over 150,000 miles now with no ill effects, I think a light weight synthetic motor oil may be fine but a standard 15-40 may be pushing your luck, especially in cold climates.
Jared