I've been on all 3 of those routes over the Bighorns with many different vehicles, from SUVs to 42' coaches towing a Jeep.
US-16 goes past Thermpolois (hot springs)and through beautiful Ten Sleep canyon on the way to Buffalo. I'd recommend that drive to ANY
RV going in either direction - eastbound or westbound.
US-14 or 14A is a different story, however.
Heading east from Cody takes you to US-14 or US-14A and climbs the steep grades on the western slope of the Bighorns up to Burgess Junction at the top. Once up top the downgrade on the east face of the Bighorns is less steep and there are plenty of turnouts to rest your RV's brakes if needed. The westface is relatively devoid of turnouts and is much steeper. US-14 is a steep climb. We've made that with our 2003 Suncruiser 33V/Workhorse and it received the workout of it's life on that climb. Pedal to the metal climbs for 15 minutes at a crack with no relief and a constant 35 MPH max speed. It's doable but if you really have to go that way I'd advise unhooking the toad and let the copilot follow behind to relieve some of the stress from your system.
US-14A is even worse. We've done that with our 40' Bus towing and it too is doable. However, the difference was that we had a 400 HP diesel pusher that could better handle the stress of the climb. I would not advise taking a gasser up US-14A. US-14A is super steep with no place to pull over. The only reason we did it the last time was because we were near Lovell looking at the wild mustang herds in the Pryor Mountains and Lovell is right at the base of the Bighorns on US-14A.
In either case I would not take any RV westbound on either 14 or 14A. There's just no place to rest the brakes. Here's a picture of the sign at the top before descending westbound on 14A:
That'll give you an idea of what it's like.
However, US-16 has less grade to it and there's lots of flat areas where you can let the coach breathe a bit. Plus there's plenty of pullovers, many of which are nice scenic spots. I'd give US-16 two thumbs up in either direction.