Hi! We have the same set up, and we have been traveling back and forth from AZ & IN to Yellowstone for 4 years. We work here during the summers! So, while we haven't traveled here from the SE, I would still recommend either coming from I 10 through Nevada and up I 15 all the way through UT & ID, and then take HWY 20 at exit 19 in Idaho Falls all the way to the west gate of the park in West Yellowstone. Madison CG is about 14 miles from the west gate.
If you go north before NV, I would get up to I 80 at least in NE, and take it all the way through Wyoming to get to I 15 in Utah and then up from there.
These aren't the quickest ways, but they keep you near diesel truck stops and off of winding mountain roads.
I 40 is another possibility, if you choose this, I would recommend taking it through Kingman and then going over the Hoover Dam and connecting up with I 15 that way.
The more travel time you allow yourself, the less stressful the trip!
The weather out west is still mild in September unless you're at high elevation. The Park averages 8000 feet elev. and we can have snow any day of the year. (we don't usually, but the opportunity is there) You won't need to winterize, but I would recommend an electric water hose like pirit or heat tape for the water hose. If it does get into freezing temps at night, just don't connect the sewer, and pull the hose out to dump as needed. Madison doesn't have hookups, so you won't have to worry about that there.
Hope this helps. - Lori
