As a veteran of two July visits to Zion, with the second visit in a tent, the heat is quite manageable. Would prefer Zion's dry heat to summers in the humid Midwest.
You don't need to worry about the Narrows hike being hot, you are in a slot canyon, hiking in a cold river. If anything, getting too cold is sometimes a problem. Also found that the lawn and shade in front of the Zion lodge is a great place to cool off.
I echo the concern of others here, you are biting off more miles than you should. I've done three monster trips of 5500, 5000, and 4500, in 4-5 weeks each. Naively thought I could cover 600+ miles in a day like in a minivan. While we did cover that distance, it was not enjoyable for anyone. By the end of the first trip, found the sweet spot to be 300 miles (no more than 6 hours driving).
Perhaps dip down into Colorado and then head back?
The national parks of southern Utah, along with the Grand Canyon are worthy of a couple weeks to themselves.