I've driven that road a great deal in 18-wheelers. If it's just as convenient for you to wait a few months, then that would be good. There are several steep hills that often are icy. There is wind that regularly tips 18-wheelers over. Rawlins to Laramie is probably the worst for wind.
It IS possible to look at the national weather map and if it is totally clear all the way from California, then it may not be so bad. The weather usually travels from the west to the east at almost driving speed. But sometime a surprise drops down from the north.
If you do go now, get a set of real chains, and practice putting them on.
A bus that is considered rust free in Salt Lake City may not be what we would consider rust free on the West Coast. SLC is definitely rust country. I bought both mine near Portland, Oregon -- totally rust free.