Hmmm... just realized that we will be in Bozeman just about the same time you are heading through. My daughter will be starting college there in the fall, so we will be moving her into the dorms within a day or so of when your family will be traveling through the town. So just a heads up... Bozeman will be a very busy town that week with all of the students returning to school for the fall semester and their families in town to get them settled in. (Bozeman & Missoula are two of the biggest "college towns" in Montana so both will be busy that week with students returning after their summer breaks.) Just thought I would warn you since I noticed our over-lapping dates travel dates. ;) Have fun on your trip.
Oh wait, I forgot... you could probably take a day trip to Cody while you are staying near Yellowstone Lake. If you take the east entrance in/out of the park, you should be able to do it pretty easily.
I also think you should keep the SUV for your rental car. Although a sedan would be fine, an SUV would be a better option. Most of the roads you would be on would be paved, but there are often gravel roads on many of the scenic detours across Montana. There is also ALWAYS a possibility of snow when you come to Montana. In that case you would probably be more comfortable in the SUV with 4 wheel drive as an option. Even the main roads and common routes have a possibility of needing the SUV. We took a trip to Glacier in late August in 2006 and while we were inside the visitor's center at the top of the Going To the Sun Road, it started snowing on us!! (Not an uncommon occurance at our high elevations.) You will also see snow at the top of the Beartooth Pass. That road is typically only open from about late May until September/early October because of the snow. However, we have already had record snowfall in Montana this year, so there will likely be even more snow than normal at the top of the Beartooth Pass. The roads should be clear, but snow is often found just a few steps from the side of the road and you are better safe than sorry.