We have done 2 trips to the Grand Canyon using the route you mentioned. Between the two trips, we used the following campgrounds, some of which may fit your timing:
Casino Queen RV Park, just east of St. Louis.
Tulsa NE/Will Rogers Downs KOA, FHU sites in a parking lot for the casino/race track. We had breakfast in the RV watching the horses having their morning exercise on the track.
Amarillo KOAGrants/Cibola Sands KOA - they have a short trail through a lava field adjacent to the campground.
Holbrook/Petrified Forest KOA - for a tour of the Painted Desert & Petrified Forest.
Meteor Crater RV Park - for a tour of Meteor Crater.
Lastly, consider driving north on route 89 and enter the Park on route 64. Route 89 passes
Sunset Crater and
Wupatki National Monuments, which are worth a side trip. Route 64 has views of the canyon of the Little Colorado River and takes you along the Grand Canyon Rim on your way to Trailer Village.