My family did a similar trip with children aged 2 and 5 in the early 1970's. We did it with a 15-foot TT which we had owned and used extensively for several years. The trip took a year to plan (no internet to help) and was the adventure of a lifetime.
My personal recommendation is a small bare-bones TT, under 20 feet. Your kids will be safer riding in the tow vehicle in their regular safety seats than they would be in a class C. At stops, you will have the TT for naps, meals, bathroom. You can park the TT and go off exploring in the TV. A rig this size will fit into the smaller sites often found in national parks.
You will have to do some pretty intensive planning to make this happen in your short timeline.
Have a wonderful trip!
