Most likely you'll get the cheapest air fares to Las Vegas but check Phoenix, also. I think it would be costly into Flagstaff and you'll have to change to a small plane to get into there. You'll just need to check prices of air fare and rentals.
Definitely stay at Trailer Village at the canyon itself. The free shuttle stops right there. Walk down the Bright Angel trail if only 1/8-1/4 mile. It will give you a very different perspective of the canyon.
You will be coming in via the south entrance off I-40. When you leave, exit via the east entrance on to Hwy 89 and head north into Utah.
With only 10-14 days try to spend two nights at each stop and you'll need one day on each end for your departure and return of the RV so you'll have about 8-12 days @ 2 nights each = 4-6 parks.
So I'd recommend Grand Canyon, Zion (Watchman campground in the park) - exit east to 89 north and the mentioned tunnels will not be an issue with any rental; 89 north to Hwy 12 east to Bryce - the park has two dry campgrounds (no hookups) or else stay at Ruby's RV at the park entrance and the free shuttle stops there to take you into the park. Try to do a hike below the rim. It will be like a fairy wonderland down there.
Continue on Hwy 12 to Capitol Reef - the campground in the park is shaded and very pleasant. No hookups. There are RV parks in nearby Torrey.
Then Utah 24 to I-70 east to 191 south to Arches and as previously noted, the campground will be closed but there are plenty of RV parks in town. If you're hikers, try to reserve ahead of time the ranger-led Fiery Furnace hike. It's awesome and there's a lot of shade for it.
If you feel you have the time then head south on 191 to Monument Valley and stay at Gouldings. Take a tour.
Then you'll be heading back to Las Vegas via Arizona 160 west to Hwy 89 south to Flagstaff and west to Las Vegas.
Don't try to cram too much into your short stay. These parks involve a lot of driving time.
You'll have a great trip even though it will be hot. People have to travel when they can and if you have no other options then enjoy! Drink LOTS of water in these desert regions.
Full-Timed for 16 Years
.... Back in S&B Again
Traveled 8 yr in a 40' 2004 Newmar Dutch Star Motorhome
& 8 yr in a 33' Travel Supreme 5th Wheel