I think our goal was to finish in about 2 months, but that'd put us in Pueblo far before 6/15. Our goal was also to camp primarily. I'm getting the sense from this thread that we should see if we can reschedule to warmer summer months.
First, two months would be a daily average of 75 miles. That may be possible if you're lightly loaded, in good shape, and perhaps on the young side. For many of us, riding that far every day for weeks on end would be a stretch. What's your training/conditioning plan?
Second, the first two weeks of May seems to be the traditional start time for westbound cyclists from Yorktown, in large part because it works out so well -- on average. Starting a couple weeks earlier than that might work well for you, especially on a somewhat more relaxed schedule. (It would have meant missing 5 days in a row of 100F afternoons in Kansas if I'd taken that option!) Of course, it might also mean frost in the Appalachians.
Lastly, SW Virginia tends to be cooler than many other parts of the state. For example, I left college (less than 10 miles from the TransAmerica route) in late March my senior year wearing my winter boots, because I'd needed them for months to navigate frozen sidewalks. Imagine my surprise in the piedmont, when I walked into an interview in boots -- it was warm and sunny over there! Now, if your imagination can stretch that far, extrapolate from 5,000' maximum altitude in Virginia to 10,000 feet in Colorado, and nearly that high in Wyoming.