As a Louisianan, I'd say this is nearly impossible with a post-1900 POD. The only way I see this even being a possibility is in the event of a break up of the US. Even then, for such an entity to exist, it'd need to incorporate New Orleans or Baton Rouge (or both) to make it economically viable. As neither of the cities (or their major suburbs) is majority "Cajun" or "Creole" and their residents would outnumber those of Cajun and Creole descent in the rest of South Louisiana, you couldn't really categorize this hypothetical country as a Cajun/Creole nation. Such a state would be less of a Quebec analog and more like an smaller version of an independent Texas with better cuisine. WWII and the explosion of mainstream American pop culture via radio and TV has brought the last few of generations of Cajuns more and more inline culturally with their neighbors in New Orleans, North Louisiana, and Texas. Heck, French is essentially dead as a language in Acadiana among people born postwar.
Bottom line is that the Cajun population of South Louisiana is too small, too poor, and too Americanized to make a go of it as an independent state.