[twindman]

Route 16 is by far the best (easiest) route thru the Big Horns. It sort of depends on your skills driving and pulling a trailer as to whether you think it is safe. I assume all newer pickups have exhaust brakes. If yours does it will make it much easier to go down the mountain. This also assumes you have a diesel, not a gas guzzler. If gas, I would be kind of concerned about driving in the mountains.
For the very easiest and safest route, take I 25 towards Casper, then US20, and then WY20 back up north. This circumnavigates the Big Horns all together. Of course in my opinion, you then miss the best scenery in WY.