Yes, Hwy 16 is the easiest route. Going from west to east you'll want to drop a gear just after crossing Powder River Pass, then you probably won't need to again until you're 10 miles out of Buffalo and run into a short series of 35 mph curves. (It'll be marked that trucks should use a lower gear.) You can at any time rather than braking, but it won't be necessary. I do it routinely, but I drive a stick shift and am constantly shifting.
I should also mention that as you come out of the Bighorns and approach Buffalo, there's still a fair bit of slope. (I think it's called 6% but doesn't look that steep.) An older couple driving an old motor home overheated their brakes on that section a few years ago and ended up inside a building in downtown Buffalo. He was killed. He must have been riding the brakes all the way down.
There will be two 20-25 mph hairpin curves on your way up, out of Ten Sleep Canyon, but you'll be climbing then so won't need to brake much if any.
As you're going over the Bighorns on your way back, after leaving Yellowstone, you should have a good grip on mountain driving by that time. Hwy 16 over the Bighorns is no more difficult than what you'll see in parts of Yellowstone. It's a good, all season highway. Take your time and enjoy the scenery.
It never hurts to stop at one of the many pull-offs to give your brakes the touch test, and while you're stopped take a picture of the scenery. If the hubs feel hot, take a 30-60 minute break.