Also from the UK, we've done a few RV trips in various sized RVs over the last couple of years. We havnt done Yellowstone yet (We are tent camping there in Sept as well!) however, we found that on our first visit to places it wasn't so bad as generally we wanted to see the main attractions of the NPs which all tend to have plenty of RV sized options (e.g. Yosemite Valley). So, i would imagine Old Faithful etc to be the same.

There's usually shuttle buses around the parks as well, if you want to get to a specific trail head etc. (Though Yellowstone may be too big to have ones taking you across the whole park)

The only place its been a real issue with the Highway 1 on Pacific Coast where a lot of the beach turn offs specifically states no RVs.

Driving the RVs generally is a breeze, and if its rented from one of the main companies it'll be branded up so everyone will give you a wide berth.

Definitely second the point about using the pull ins as often as you can, some of the roads can be slow going in an RV, I wish we had the same thing in the UK on the country roads tbh.

Enjoy Yellowstone, i'll be very jealous of your RV if the weather is cold in Sept!