What a neat place to run!
I did almost the exact same route (only hiking, not running) in reverse about 35 years ago. Down South Kaibab, up Bright Angel. And it was in May, not July. I can imagine it must have been sweltering. I remember my quads were screaming by the end.
Wow! In July! I've been there twice, in January and November, and those were plenty warm enough for me.