It is true that the bass side will not harmonize properly a good part of the time.

As others have said, that bit of dissonance is part of what makes Cajun music what it is.

The best thing to do is to think of the bass side as part of the rhythmic accompaniment (like a drum), and not as harmonic accompaniment (like a rhythm guitar).

Just keep the rhythm going with the bass side, at least if you are playing in first or second position (C or G, on a C box). If you are in third position (F on a C box), the bass side is often left out because it gets in the way. Same with Pine Grove Blues or Danse de Mardi Gras.