Starting in June?!?
The heat bothered me in Kansas in June a couple years ago. The combination of long days plus heat just about wore me out. I've worked outside in the New Mexico desert for a few days, and it's even hotter there. While training in Florida, do you normally ride 5-8 hours daily for weeks, with only an occasional day off? That's what you're talking about doing. I'd consider a month of southern California through Texas in the summer to be, quite literally, life-threatening.
Touring, and riding daily, means you have to be really careful about water and electrolytes. If you decide to go there, then, be very careful about packing and drinking enough water, and pull a trailer to tote all the Endurolytes you'll need.
And with all due respect to the weather channel, I'm not convinced this summer has been all that unusual. Any place I go, any time I go, the weather seems to be abnormally hot, cold, rainy, dry, etc. The Southern Tier goes where it's usually hot in the summer. If it's 108F in Arizona, I doubt most people could stay outside for four hours without a thermometer and swear it wasn't 112.
Sorry if I'm rambling. This sounds like such a bad idea I don't know if I can be any more coherent. I'd go north. Either to the TransAm or Northern Tier. It'll still be hot, but not quite as hot; it'll be cooler in the mornings; and you'll occasionally get a break in the heat.