I don't think it is a great idea.
I've ridden the Southern Tier in late winter.
I wouldn't want to consider it in mid summer.
Since it's your first post there's not much to go on.
Here's what I am guessing:
You want to ride cross-country - -
But you only have 45 days to do it.
And it has to be in the May 20 to July 4 window.
Plus you are not familiar with desert Southwest conditions.
Now, that is not an ideal combination if you step back.
Yes, you could probably get through it.
But is that what you want - simply to get through it?
Heading west to east on the ST -
You are in the Imperial and Sonoran Deserts by Day 2.
Plus June is the hottest month in much of the Southwest.
Then you add highest temps and humidities of the Southeast.
It borders on the danerous - plus you leave yourself no leeway.
I agree with my colleague that the TransAm/Western Express combo is better.
A fast westbound trip starting in Yorktown would be in good weather, mostly.
YMMV - weather normals are long-term averages.
But the TA/WE is 3800 miles vs the ST at 3000. (Let's say 43 days.)
That's the difference between 70 miles/day and 88 miles/day.
(Frankly, 70 mpd on the ST in summer will be pushing it.)
I see a number of possible options.
First, you could add a week and do the TA/WE fast, but reasonably.
Second, you could map your own more direct route - say LAX to DC.
That would be perhaps only 3200 miles.
(See Tzuo Han Law's route - http://www.crazyguyonabike.com/doc/?o=1&doc_id=2405&v=2LH
I helped him with his route - he got up to 100 miles per day.
It includes an easy start with gradual climbs out of Ventura.
Then involves stretches of Historic Route 66 and the Grand Canyon.
You pick up the TransAm in Colorado and stay on it until SW Virginia.
Then take the Blue Ridge Parkway / Skyline Drive towards DC.
But the Southern Tier for those dates?
There's better riding out there.