In 2011 we did something similar to what the OP proposes. I had a week-long professional conference in Vegas over the Labor Day weekend, and dragged our TT to Vegas for the conference. At the end of that, my bride flew out to Vegas, and we dragged the trailer through Kingman to the South Rim, staying for several days in Trailer Village. The way to get around on the South Rim is via the free shuttle bus system, and we made use of it to explore the South Rim.
We then dragged the TT around the east end of the Canyon and spent a couple more days enjoying the North Rim, which is a very different place. We then went west on 89 to Bryce, then to Zion, before returning to Vegas, where my bride caught another flight home. I dragged the TT back home by myself.
The journey from Vegas to the South Rim is a one-day trip. You MUST have reservations in Trailer Village, although there are campgrounds just outside the park in Tusayan where you might not need them.
We did not hit other places, except Kodachrome State Park, on our journey, which was limited to two weeks by the vacation time my wife could get. It should be noted that we like to go somewhere and explore a bit, rather than rushing from one attraction to the next as others often do. If you have only two weeks, the number of sights you will be able to take in will depend upon your travel style. Given your proposed itinerary, I think you would do well just flying into Vegas and renting there.
Do be aware that entering Zion via the eastern park entrance will take you through a tunnel with height restrictions that not all rental RVs can pass. In case yours doesn't, you have to circle around and enter the park from the west.