My guess would be that KY, TN, VA and maybe NC, would be a little more liberal, because liberal ideas from the US could seep south.
I might throw Georgia in that list as well. Pre-Civil War, the (USA) Whig Party was fairly strong there, so Georgia seems to be a little more "liberal" than the rest of the Deep South. And it includes Atlanta (I assume the Rad Libs would be stronger in the larger cities.)
And even Alabama might be a possibility. Afterall OTL it elected Oscar Underwood to the Senate around this time, and he opposed both Prohibition and the Ku Klux Klan. (It wouldn't surprise me at all if Underwood was a Radical Liberal in TL-191.) The fact that Alabama elected Underwood suggests its people might be more liberal than we might expect for a Deep South state. (And of course in TL-191 they have Birmingham as a major center of Confederate industry, so if the Rad Libs dp run strong in the cities that would give them an important stronghold in Alabama.)
Both Alabama and Georgia would be tough wins for the Rad Libs. They probably couldn't carry either state in a one on fight against the Whigs or Freedomites, but with the right wing vote fractured between those two groups they might just have been able to pull it off.
Trotsky said:
The Radical Liberals were seen as the party of "surrender" by everyone else. Sucking up to damnyankees was a very unpopular thing in the postwar scene.
Most of our white Confederate POV characters are hard core Freedomites, not exactly an unbiased viewpoint. Reggie Bartlett is the closest thing we had to a Confederate everyman, and he certainly didn't view the Rad Libs as the party of surrender. And even Clarence Potter seemed to respect them. He was foremost a Confederate patriot, so he would not have had any good thoughts about them if he thought they were the party of surrender.
As for sucking up to the damnyankees, as with most things in politics it depends on how you spin it. If the Rad Libs make the case that they can use diplomatic means to convince the US to end the reparations that are destroying the Confederate economy, I think that idea would be very popular with the Confederate people.