Even if you get to the south rim, at 6800' you may find significant snow that makes it tough to walk out to some of the view points.
We've camped in the Tucson area in the time frame you are talking about and had a great time. But as we headed home, conditions got nasty. Now I'm a CO native, drive to go skiing all the time, and have camped with a trailer for 20 years, and it wasn't a fun drive once we got north of Phoenix.
Around Tucson you can visit the various parts of Saguaro NP and head over to Kartchner Caverns. Near Phoenix there are a couple of NM's preserving native ruins.
Another thing to note is that even in the summer, the limited amount of time your have would make your trip from Phoenix to Denver pretty rushed.
