My two cents is this seems both foolish and dangerous. High temperatures in the southwest deserts average above 40C and do not cool much in the afternoon.
Everyone body works differently with heat but I found for myself that more extended riding above body temperature (37C) was tough and also got heat exhaustion in Sudan on a day it got to 47C.
For that time of year, I suggest perhaps a different "crossing" of America such as from Canada to Mexico along the Pacific Coast or perhaps a ride from Pacific Coast to Chicago this year and finish the job on another ride.
Two general issues with the ride early scheme (a) you'll want to start earlier probably in dark and finish ride by 11am (b) you'll want some place out of the heat for the afternoon into the evening; some places motels ($) and a few spots have longer gaps between accommodations. Adventure Cycling has a route along much of your path and here is the page they describe for when to go -
https://www.adventurecycling.org/blo...southern-tier/