Assuming that you are able to use the Tesla Superchargers in a rental car — not sure how that works — You can easily get to Mesa Verde via the Supercharger network:
• Denver (starting with a 90% charge, I presume) to Poncha Springs Supercharger Station, charge enough to get to Farmington with 20% or so. Distance from the Denver Supercharger Station near (sort of) the airport to Poncha Springs: 155 miles, about 131 "rated miles."
• Poncha Springs to Farmington NM Supercharger Station, 230 miles, about 170 "rated miles," Take your time and charge to 90% at Farmington unless you plan to charge in the park.
• Farmington to Mesa Verde National Park: 73 miles to the entrance station. Be aware that it is a long way from the entrance station to the main part of the park so you want enough charge to get into the park, do some exploring, and get back to Farmington to charge again and return to Denver (if that's the plan). This is fairly easy to do in a Model S with a battery at 250 "rated miles."
This is an easy one day drive each way. However, it is a trip that will take several days in order to get to and visit the park — you would need at least one day in the park to explore at least some of the sites. So, where do you want to stay? Hotels? Camping? There are campgrounds with RV pedestals that will allow you to charge the Model S (be sure the charge cable is in the car before leaving Denver, regardless of whether or not you plan to use it). There is also a Tesla Destination charger at Far View Lodge
inside the park if you choose to stay there:
Far View Lodge | Mesa Verde National Park CO | VisitMesaVerde.com
If staying at Far View Lodge, let them know that you will want to use the Tesla charge station at the hotel. There is also a J1772 charge station that can be used by any EV, and by a Tesla with an adapter (again, be sure this adapter is in the car), in case the Tesla charge station is in use.
Because the main attractions in the park are so far from the entrance station, it is best to stay in the area, to allow more time to explore. Far View Visitors Center is about 15 miles from the entrance. The museum and Spruce Tree House is about six miles from Far View. These are slow twisting roads with significant elevation gain (however, the slow speeds mean that you will use only a tiny bit of the car's charge, so it isn't a range issue
at all). Yes, you could do an out and back from Durango but that would be a long day and would limit the time exploring the park. If you do get all the way into the park, do check out the little museum next to Spruce Tree House — it has original artifacts recovered from the ruins.
Is the plan to return to Denver the same way you got to Mesa Verde? How much time do you have? Another alternative would be to use the Supercharger network to go to Blanding UT, Moab UT, and check out the parks there. You could do a loop and head back to Denver via I-70. There are many other loop alternatives that could be tried. Depends on the time available and what you want to see/drive.
September could be hot, or rather cold, at the altitude of Mesa Verde NP, about 8000 feet. Bring clothes suitable for cool weather, just in case.