We live in Kansas City, so have some info on climate and routes.
Understand that you will be in the humidity-filled Ozark geological region across the entire state of MO and into Eastern KS. Once you approach a N/S line that runs west of about Manhattan KS you will be into the Tall Grass Prairie region, and the humidity will be much less. It will still be HOT. Can be 95 degrees every day. But in MO it will be that hot AND be humid.
So traveling you will be in shorts/sandals. You will not be able to leave a pet in a vehicle if you go eat or shop. You will want electric at night to run the AC so you will be able to sleep.
If Columbia MO is a good stopping point, look at Cottonwood Falls RV park. Very nice and good location convenient to the highway.
If you like state parks, MO has electric but most don't have water at each campsite. Weston Bend or Lewis and Clark are in the general KC region. Reservations needed. Make on MoStateParks.com
https://mostateparks.com/campgrounds...end-state-park
There are some Corps of Engineer parks also not far off your route. These will have water at each campsite. We like Melvern-The Outlet. Reservations needed. Make on Recreation.gov
There is a nice KOA in Limon CO if you are ready for an overnight stop there. Usually we can sleep with windows open by the time we hit CO, but it will still be 85 during the day. It just gets cooler at night in the Desert climate.
When in the Denver metro, for the love of all that is logical, take the tollway around. They mail you a bill. Denver traffic is terrible, and you will need/appreciate all the help you can get.
If you do want to stay in Denver, look at Chatfield State Park. Reservations needed. If you want to go into the mountains, then you may already have trouble getting reservations unless you are OK to get campsites without electric. Recreation.gov for areas around Frisco/Brekenridge/Dillon Lake.