That should be a great time to make that trip, although the North Rim gets dicey weather-wise and generally closes for overnights by Oct. 15. We are making basically the same 2 wk trip, leaving from Northern IL, in mid June. Check back, as I'll try to make a full trip report. We will be staying at state or natl parks. I would be willing to try some of the BLM spots, but that time of year I want to at least have electric for AC (kind of locked into June because of kids school).
We plan to first stop at Rifle Gap SP in CO for a few days, to do a few things in the Glenwood Springs area. Then to Moab, where we will stay at Dead Horse Point SP for a few days and visit Arches NP. Then on to Bryce, but staying at Kodachrome Basin SP, because it has full hookups and seems better suited to a 36' RV than the Bryce Canyon NP campgrounds. Finally, four nights at Zion NP in the Watchman CG. We thought about the North Rim, but decided that would be adding too much to the trip. If you want to stay in any of the NP CGs and want to have reservations, start thinking about it soon. The NP reservation window starts 6 mos. in advance. The Utah SP reservation window is four months in advance. I suppose Sept. will be a little more off-season, so you might not have the same reservation issues that I had, but when I tried to make a reservation at DHP SP 15 minutes after the reservation window opened, the weekend nights were already all full.
A couple of tools that I used extensively in planning/dreaming were the AllStays "Camp and RV" app for tablet or smartphone, rvparkreviews.com, and campsitephotos.com, which shows photos of the actual campsites at various parks. The geographic coverage at campsitephotos.com is pretty spotty, but they seem to have most of the Utah SPs and NPs.
I imagine others with much more experience (notably, SeilerBird) will chime in soon with better advice. [Doh, he already beat me to it, I see when I hit "post"]