I'll give you my personal opinion. We drove through Yellowstone several times, leaving our trailer in Cody. I would not have been easy in my mind pulling the fifth wheel through Yellowstone. The traffic can be brutal at times. When the wildlife is abundant, people will pull over on the side of the road to get out and take pictures.....on both sides of the road and that leaves about a lane-and-a-half for two-way traffic to get through, sort of like threading a tiny-eyed needle with a piece of twine. If you wanted/needed to stop for any reason, it'd be the devil to find a parking place for an RV/car combo.
My recommendation is to go north on 310, west on 90 and south on 356. The mileage through the park from Cody to West Yellowstone is approx. 135 miles. The alternate route is 356 miles, but you can go faster. At the best conditions, it takes about 3 hours to go through the park (and if you're there in tourist season, there are seldom "best" conditions) or 6 hours on my recommended route. If there is a major "snag" in the park, you're pretty much stuck there.
We've been north on 310...no problems. West on 90 is absolutely no problems. I've never been south on 356, so I can't give a personal recommendation, but it is the main route from 90 into West Yellowstone, so it can't be too bad. Going south on 191 from 90 is an alternative, but I don't think it's as good a road as 356 and the time works out about the same. You can also take 89 south from 90 to 191 west (it cuts through a corner of the park) but I think it's a pretty narrow road, too.
Hope this helps. Have a great time, however you go! Where are you camping in Cody? We stay at the Ponderosa - great location.