If yo are going on hiway 14 you should have no problem as long as you know how to drive in the mountains. Always use the truckers recommendation of always going down a hill in one gear lower than you went up it. 14A is a different story.
My wife and I have gone down 14A several times headed west. We have towed popups pulled by both Chevy Caprice and Mercury Grand Marquis. We also went down 14A going west with a Toyota Escaper w/4 cylinder and auto. I have never had hot brakes (or even warm brakes) on these trips. The thing to remember is that you always go down a hill in the gear lower than you would use going up. I frequently end up going down extreme hills in first gear and use my brakes as little as possible.
This is a quote from the Wyoming Mountain Road website--
"Highway 14A is extremely steep, a 10-percent grade and more, is winding, has several hairpin turns, and many blind corners. This is a challenging road during the summer season and closed during the winter.
This road is not recommended for oversize or over-length loads. Unless you are experienced in extreme mountain road driving, it is not advisable for 18-wheelers.
For tourists, it is a beautiful drive with wildlife, wonderful scenery and over 200 mile views in some areas. However, if you are driving a motor home, please pay attention to the information in the above paragraphs. Use lower gears and save the dangerous possibility of burning out your brakes."
Even after reading this, I will continue to go through the Bighorns when I travel. If YOU are nervous about taking this route, take hiway 14 instead.
