Summertime strong winds are usually associated with thunderstorms. It's impossible to predict them... anywhere.
There's no "Bighorn National Park," only a Bighorn National Forest. This is a guess. You're planning Hwy 16 over the Bighorns? If so, that's the one I usually recommend. It's a mountain road, but it's a good, paved, wide highway with mostly gentle curves and not-too-steep grades -- if you're careful and don't ride the brakes.
(Use lower gears and use the brakes wisely. Don't "ride" them, don't "tap" them. Brake hard to slow, then get off of them until you must use them again to slow. With an engine brake you won't need the brakes much. If you find that you're using them much, pull over and do the touch test on all wheels. If they're hot, take an hour break and enjoy a picnic or some roadside photography.)
The mountain driving on Hwy 16 will be between Buffalo and Ten Sleep (60 miles?) You'll get into it again once inside Yellowstone -- nothing that all rigs can't handle; just take your time and use your noggin.
