Although this may be a little late, there is a lot of great Cajun music in Eunice and surrounding areas almost every day between Friday and Tuesday. Just go there>
As for the Balfa Brothers recordings of the Mardi Gras Song, they have a good number of recordings of the song on different LPs and CDs. I am not sure which specific version is the one that you are looking for, but I think that it may be one of these;
1. "Cajun Fiddle Old and New," on Folkways FM 8362, released in 1977, Track 11. I think that Allie Young plays accordian in that LP, though.
2. "Nathan Abshire and the Pine Leaf Boys," Volume 13 of the Legendary Jay Miller Sessions, on Flyright LP 535, released in 1978, Track 7.
One or both of these versions may be on Neal Pomea's website. If not, you can get the Folkways recording from the Smithsonian Folkways website. When Moses Asch agreed to have his Folkways recordings transferred to the Smithsonian, it was with the proviso that all of the recordings would remain available forever. The will make you a custom CD of the LP.
The Flyright recording might be a little more difficult to find. Several years ago, I know that Mark Miller in Crowley still had copies of the LP in the attic of his store. He may have also released it on CD.