Is there some reason you aren't taking Hwy 12 which is the normal route. It does have some steep grades but also spectacular scenery. We did it without any problem pulling our 34 ft fifth wheel and there were plenty of other big RVs on the road with us. If you are underpowered then that could be a problem tho.
62 is most likely fine, I don't have any specific recollections of that road but over the years we've been up and down most of them in the area.