I've been to Yellowstone many, many times, spring, summer and fall. I think my favorite time is early June. The babies are out playing then and crowds don't arrive until late June. Late May is okay, but it's still pretty cool and wet in May, and every week you can put it off the better it gets until the crowds descend.
I've towed 5th wheels over Hwy 12/Lolo a few times. It's a beautiful drive, but it's slow going -- lots and lots of 30-35 mph curves, almost continual. My biggest concern is bicycles on the road. I nearly hit a couple of them the last time -- coming around a curve and there they were, peddling about 5 mph with no shoulder and I was meeting another camper. I slammed on my brakes and was able to get stopped (or slowed to 5-10, but it could have been ugly. The highway gets lots of bicyclists, and there's often very little shoulder for them, so beware if you go that way.
West Yellowstone, MT would be my pick if I wanted to stay outside the Park. It's right AT the west entrance, and the road from there to the main loop is a pleasant drive. If you find the traffic to be heavy going in, leave earlier in the morning. Yellowstone is best seen just after sunrise. Even during the height of tourist season, the roads are empty then. It's also the prettiest time of the day, especially for photographs.