New Orleans is not the only Catholic centre in Louisiana. The strength of the Catholic Church is very tangible in Louisiana, both in the Archbishop seat of New Orleans, and into the parishes. No Protestant government in Richmond would dare step upon it.
Coastal Mississippi, Coastal Alabama, Charleston and Savannah also had relatively large number of Catholics.
I don't think a Protestant government of the CSA confederation would be inclined to harass or suppress Catholics. Sure, there was some religious animosity from various preachers with small followings. Historically, however, the Confederate leadership identified with and respected the
ancien regime aspect of the Catholic church as they felt it encouraged traditional social arrangements and traditional culture. Of course, "traditional" to the CSA meant "Planter Culture"
In its second wave the KKK was very hostile towards Catholics and Jews.
Most of that was hostility to immigrants (of whom a strong majority were also Catholic). Though there was a faction of the KKK inherently hostile to Catholics as a religion, most of the KKK divided Catholics into "indigenous" and "immigrant"- then saved the hostility for the immigrants- especially those with non anglo saxon appearances.