It's all about the sugar. I don't know the specifics of trade policy back then, but a bull moose candidate actually got elected in that area in 1918 (he switched to the D to ensure he could stay in office, but the politics of sugar remained)
The main reason for the 1920 bolt was not exclusively the sugar issue, but equally or more that Woodrow Wilson was perceived as anti-French because of his bitter disagreement with Georges Benjamin Clemenceau over war and postwar policy. Wilson did not believe France was entitled to the Rhineland, Saar and Ruhr as Clemenceau did, and Cajun Louisiana sympathised strongly with Clemenceau.
In contrast, during World War II this region’s pro-France and anti-German sympathies caused it to behave like northern New England.
1920
Now here it's where it gets a little funky. As you can see, Harding won quite a few parishes in Cajun country. I have no clue as to why; Wikipedia claims that Cajuns were offended by Woodrow Wilson feuding with Georges Clemenceau, but frankly that seems kind of a weird result unless there was some more overt Francophobia or Catholic hating going on in the Wilson administration or the Cox campaign – at least more so than was usual for the time. It doesn't seem crazy to think that a political party with close ties to the KKK would have trouble winning Catholic voters. Harding didn't win all of Acadiana – presumably there were some Democratic machines, like Leander Perez's in Plaquemines Parish, which stopped anyone from voting for Republicans.
As I said before, there was no need for Wilson’s Francophobia to be overt – for the Cajuns it was enough that it was perceived Wilson did not agree with French demands and they would refuse to support him.
There are so many other issues of the Wilson administration, notably Prohibition and the 19th Amendment, which the devoutly Catholic Cajun parishes would have been extremely hostile to. Moreover, as they were less obsessed with racial issues as the rest of the Deep South, they felt they could afford the luxury of voting their views on these issues as other Southerners could not.