The Kennedy brothers produce Neo-Confederate history texts such as "The South was Right!" (https://books.google.com/books/about/The_South_was_Right.html?id=QFilZzg3GxkC).
How seriously are they taken? I'm in a couple of American History classes at university and the Lost Cause ideology seems to be scoffed at by modern historians. Professor Brian Dirck at Anderson University reviewed "Was Jefferson Davis was Right!" and seemed to sum it by saying it was "ham-handed Neo-Confederate polemic." (http://www.jstor.org.offcampus.lib.washington.edu/stable/2567108)
So, what's modern American academia's take on traditional conservative Southern Civil War history?
No, not in the slightest. These guys (neo-Confederate activists/writers) exist far outside the mainstream of serious historical discourse; I'm frankly a little uncomfortable even lumping them in with "traditional conservative southern history." Shelby Foote may have had some biases, but he stands head and shoulders above this crowd. What they are, basically, are fanatics and political hacks: propagandists and polemicists for a cause, not scholars. They don't write for scholars, and they don't honestly read the works of reputable scholars. Rather, they manipulate primary sources and ancient secondary sources to produce wildly biased narratives intended for consumption by a historically illiterate and credulous lay audience. The end goal is rebuilding Confederate nationalism (or at least stoking sectional tensions), not arriving at any kind of a nuanced or comprehensive view of the Civil War.
No, they're not considered good academics in the slightest.
However, in the academy, there are journals that are pay to play, so you can pay to publish your work. It's not going to be peer reviewed and it's not going to be quality, but you're published. That's how you can end up on sites like Jstor, even though your work is shoddy.
And I am sure you know you can publish whatever books you like.
Now, I've not read the Kennedy brothers' work so I can't critique any of its claims, but saying "the South was right" is not really an academic argument. To me, it's a bit like saying, "Hitler was right." I mean, if you're a fucking racist, sure, maybe he was right. If you liked slavery then yeah, I guess the South had a valid point. But, again, that's not for historians to say.
Generally, historians just examine people/events--they don't go around publishing books titled, "LBJ was a great president!"
By the way, the granddaddy work of modern Southern history is C. Vann Woodward's Origins of the New South, published in 1951. He was a Southerner and a brilliant academic and the book's still being taught and I recommend it. Basically, Woodward makes it clear that whatever problems the South had before/during/after the war were the fault of Southern politicians and their policies.
In the vein of the original post, there does seem to be a small academic subset of Southern professors who are promoting scholarship in defense of the South. A professor from Emory University started the Abbeville Institute as a safe space for defending secessionism, the Lost Cause and Southern virtue.
https://www.chronicle.com/article/Secretive-Scholars-of-the-Old/49337