In the desert June isn't any better than July. However, monsoons start in July and rains can wreck havoc in the deserts so I'd recommend June. You're kind of late for reservations for this summer because you're on a tight timeframe so perhaps next summer will be best and get your reservations early.
Each national park has a great web site that gives you lots of good information on things to do, camping, road information and special alerts. If you check out Arches site you'll see that the campground will be closed all summer of this year for renovations and road work. That park will be a zoo this summer so if you can swing it, take your trip in June of next year.
https://www.nps.gov/arch/index.htmCanyonlands can be done from the Moab/Arches area unless you want to do the Needles section a little further south and it's completely different in looks. That campground is very, very nice and the trails are fun - walking on slickrock (not really slick). It's by Monticello. The Canyonlands section by Moab is red sandstone as is Arches. Dead Horse Point State Park is at Canyonlands by Arches. It's very nice and there are some electric sites. Fill with water at your previous stop before coming in. For Arches, try to reserve a spot beforehand for the Fiery Furnace ranger-led hike. It's a unique experience climbing among the hoodoos and it's somewhat cool (temps). Check to see if there's an age limit for the children but they would love the hike. It's very popular.
https://stateparks.utah.gov/parks/dead-horse/Here are some BLM campgrounds and the ones on the Colorado River would be the coolest - lots of trees. We also like Ken's Lake but unless you can get one of the rear sites with trees, it would be in the sun. Plan to take an evening boat ride on the Colorado. It'll cool you down.
http://www.discovermoab.com/campgrounds_blm.htmA gorgeous drive from Moab/Arches is to take Utah 24 off I-70 taking you to Torrey. Capitol Reef Nat'l Park is there and the campground is lovely with an orchard and big old trees. Then continue on Hwy 12 to Bryce. There are 2 campgrounds in the park and both are very nice. North campground has bigger sites - I don't know what you have. This will be your coolest stop (temp).
Check out Zion's web site for the tunnel drive - Hwy 9 off 89. There are restrictions and it will be a curvy, steep downhill if that would concern you. Many RVers do it in all sizes of RVs. It just depends on your confidence and confidence in your vehicle. Downshift before you even start going down and you'll be fine and you'll use your brakes much less. Watchman campground w/electric is the place to stay. It does have trees so you won't be out in full sun on most sites. If you have a smaller RV the other campground (no electric)is even shadier. The park entrance is at Springdale - a neat little town for an ice cream or pizza. The free shuttle makes a few stops there from the park or you can even walk from the campgrounds. Excellent sites in Watchman are B Loop 50, 54, 44. They're on the shallow river and good fun for the kids (& you) to cool off.
If you don't want to do the tunnel drive, from Bryce on Hwy 12 go west to 89 and then a little north through Panguitch and pick up Hwy 20 which will take you to I-15. Then south on I-15 and come in on Hwy 9 from west to east. Don't take Hwy 14 to I-15. You definitely won't like it - steep.
From Zion you will be so close to the North Rim of the Grand Canyon if you have any time left just for a look-see of one night. Again, if you don't want to do Zion's tunnels then leave Zion via Springdale and go west on Hwy 9 to Hurricane. Then pick up UT59 which becomes AZ 389 taking you to Fredonia, AZ. Then head south on 67 to the North Rim. If you can't get a site there try just outside the park gate at DeMotte national forest campground. It's right on the paved 67 road and it's very nice. If not there then the first good gravel road south of DeMotte has some great boondocking spots if you just drive in 1/2 - 1 miles.
Hope you can make this trip or at least part of it. You and the kids will really enjoy it. Even if you can't spend a lot of time in the parks, at least you can see them in the time you have. It's better than not seeing them at all!