I know a few things about this. There are several ways to skin this cat. They are all right and they are all wrong.
First, the weather.
Along the Gulf Coast states, you can expect high humidity and hot temperatures beginning in early May and lasting at least through September. Now I know other states can claim higher temperatures, and I really don't care who's pecker is the biggest, but I can tell you that until you have ridden in full gear under an honest 90 degrees COUPLED with an honest 65% humidity, just sit in the corner and suck your thumb. The States within the Appalachian range (north Georgia all the way up to West Virgina) will be tolerable further into the early and late summer. OTOH, you probably won't want to ride thee December through March. Which leaves Arkansas, Tennessee, and Kentucky, which I think find their sweet spot somewhere in the middle. Arkansas is nice through April. Tennessee and Kentucky can be very nice in early September.
To summarize, I would make this a late Summer/Fall trip, and I would approach this in a "snake" pattern if I had time. From you, I'd probably go clockwise. I'd head straight to West Virgina and try to get there in early to mid September. You could easily spend two weeks running basically parallel to the Blue Ridge Parkway. I would spend a little time actually on the parkway, but I'd spend 97% of my time on the wonderful roads that surround the parkway. When you get to North Georgia, take a right and check out Northern Alabama. Then turn right again and check out Eastern Tennessee and southeastern Kentucky. Then take a U-turn and check out northern and central Mississippi. Next, run through my State (please call me), and head north to Arkansas. Now turn left in southern Missouri and find your way home.
This can be 2 - 6 weeks total. It could be 3,000 or 7,000 miles too.
Of course you have seen my ride reports and they might give you some ideas too.
Finally, I would be insulted if you didn't give me a chance to host you, or at least join you for some of these miles. If Louisiana is in your cards, I would be honored to show you around.