You don't mention the type of RV you have, but, as you know, many national parks have few, if any, long campsites. We were just in Big Bend NP in February and the weather was great at that time of year. We skipped them this trip, but Guadalupe Mtns. NP and Carlsbad Caverns NP had very good weather at that time as well. So, the weather for that part of your route should be no problem.
We were in Great Basin NP in June 2015 and, surprise!, it was hot. However, the campgrounds inside the park are at elevation on the side of 13,000-ft. Wheeler Peak, so cold weather (and no water) could be a problem in fall and winter. Incidentally, we were on our way back to Denver on "The Loneliest Highway in America" (US 50) after visiting Yosemite, Sequoia, and Kings Canyon NPs when we stopped for a night at Great Basin NP.
I have lived in Colorado for 54 years and every fall has been different. Some years, Trail Ridge Road through Rocky Mountain NP is closed by mid-September and other years not until early to mid-October. Of course, the most spectacular time of year is about the 2nd week in September when all the aspen are turning and the elk are rutting along Fall River Road. But that is also when Estes Park traffic is at its worst.
Great Sand Dunes NP, Black Canyon of the Gunnison NP, and Mesa Verde NP generally have fine weather in September and early October, but can iffy after that. Not that snow on top of Black Canyon or on the dunes is all bad--it makes for some great photos. But you really need to be prepared for cold weather. Fortunately, Black Canyon has some electric-only campsites, so running the furnace is not a problem.
When you leave Black Canyon, think about taking US 550 to Ridgway, then CO 62 and CO 145 to Cortez. You will get to see the most-photographed view of the San Juan Mtns. from Dallas Divide (especially when the aspen are turning about the 3rd-4th week of September), take a short side trip to see spectacular Bridal Veil Falls at Telluride, cross Lizard Head Pass, and end up just a couple of miles from a highly-recommended campground for your visit to Mesa Verde.
At Mesa Verde, try staying at the Juniper Campground at McPhee Reservoir near Dolores instead of inside the NP at Morefield Park Campground. Juniper has long, level electric sites, is only 10 miles from the NP, and is right next door to the wonderful Anasazi Museum, which IMHO is much better than the museum inside the park. It also has much better access to the Canyons of the Ancients NM, if you are at all interested in more Anasazi ruins.
However, you decide to go, you can't help but have a wonderful trip!