Dunraven Pass is the only way to get from the north part of Yellowstone to the south loop, which is where most of the thermal activity is (including Old Faithful) without going thru major road construction; the road from Norris to Mammoth is being literally moved in some spots, so this "surgery" will be going on all season this year, and probably into next year. Dunraven isn't too bad - though I will admit we haven't towed over it. Take it slow & easy and the driver will have to resist the temptation to watch the scenery - and you'll be fine.
Another way to get into the park from Billings is to take I-90 west a short distance to Laurel MT, and take the exit for US 212/310. Follow US 310 south to state route 72, just past Bridger; that road will turn into Wyoming state route 120. Follow that into Cody. No high passes, just rolling countryside, quite scenic but not a difficult drive. From Cody you can drive west into Yellowstone; it's a good 1 1/2 -2 hours from Cody to Fishing Bridge.
Bridge Bay or Canyon are good centrally-located CGs inside Yellowstone. Bear in mind they do not have any hookups. If you need hookups, Fishing Bridge RV Park is your only choice within the park. As previously mentioned, you will need to make reservations as soon as possible - and it's possible they may already be booked up for your timeframe. Alternatives would be to camp between Cody and the park (which will entail quite a bit of extra driving) or on the west side of the park - Yellowstone Grizzly RV Park in West Yellowstone is good, and close to the park entrance.
