Chief Joseph Pass, near Lost Trail Pass between Salmon, ID, and Darby, MT, is just fine for our coaches; been there, done that many times over both passes. The roads may wind some but are wide and easily maneuvered, and the gradual grades aren't extraordinarily steep, and have passing lanes available.
All that said, those passes are a good deal northwest of Yellowstone. We've taken our old motorhome through Yellowstone, but that was 16 years ago and it was a bit smaller rig than our Beaver. Nevertheless, as long as you reserve campsites ahead of time so you have a base of operation, taking the coach on the main roads shouldn't be an issue, and do your main touring in your toad, a luxury we didn't have on that trip.
If you are still concerned, take an RV park site in Cody and toad the park to check out things ahead of time. We've also come in from Alpine, to the south, in the old coach. But I'm sure the Beaver would do fine. My issue with the area was expensive private RV parks and inadequate parking in Grand Teton NP for large RVs at some view points and visitor centers... and sans a toad we were only 30 ft long. They allowed cars to park in large vehicle spots with no enforcement. And like I mentioned, in midsummer you will want to plan ahead, and reserve camping spots in advance, especially in park campgrounds if you will use them; any sites, especially big ones, will be at a premium. Check this out with the Park Service months in advance to know what you'll be up against.
Otherwise, if you just want to stay in Cody or Jackson and then pass through without camping to West Yellowstone, the main park roads are no problem. You can stop at Old Faithful easily on the way through, as there is a super large parking lot. The only imposition might be when people stop in narrower road areas to gawk at wildlife, or if crews are working on a road section. That is often the case with the road north from Madison junction to Gardiner, but you don't sound like you need to hit that section in the coach.
Joel
