That depends on what you mean by desirable. There are three practical routes across the Big Horns, and a few unpaved routes as well. I have towed my 5th wheel over all three of the paved routes with no trouble, and I'm a flatlander from Illinois.
Route 16 is the least steep, but also the least scenic. the only time I ever had any difficulty there, was last September during a snow storm. See the first photo below.
Route 14 is a little steeper, but is far more scenic. If you go that way, the short trail to the falls at the Shell Canyon visitor center will be well worth a stop. This will be an easy drive going up, and slightly more difficult going down.
My favorite it the combination or routes 14A and 14. Route 14A can be a bit hairy going west, bot you wont have that problem. If you do go that way, Bighorn Canyon National recreation Area east of Lovell is an under appreciated gem. It has fantastic scenery, and both big horn sheep and wild horses are often found right along the road.
Joel