Have you considered south to north?
I have suggested that for years - although ACA has it north to south.
1. Still cool and rainy in north in June - plus much snow remains unmelted
2. Summer monsoon season begins in late June in south turning dirt roads into muck
3. Winds tend to be southerly - although in the mountains they are highly variable
4. You generally have the sun behind you - not in your eyes.
You could flip your initerary - fly into LA and take Amtrak to Tucson
Although El Paso is close to Antelope Wells with a lightly used road between
I would, however, caution against stealth camping on the US/Mex border.
Since it only takes about two months, max, to ride to the Canadian border -
Why not continue on thru the Canadian parks to Jasper - really spectacular.
(Unlike US national parks, Cdn parks have fireroads into the backcountry that permit bikes.)
I would guess that you would have enough time to ride on from Jasper to Prince Rupert on the coast -
Then take the ferry through the spectacular Inside Passage back to Vancouver Island -
With enough remaining time you could ride Vancouver or bus it to Vancouver airport.
Photo - Backcountry in Jasper National Park by Bicycle