I did a survey of the historiography of Louisiana regiments during the Civil War, and consistently, while francophone units existed, the Acadian Guard of the 18th Infantry being a notable example, there was never any mention of linguistic barriers. Nearly every Louisiana unit (and one East Texas cavalry unit) was absorbed into other units as the war progressed and casualties mounted. There's Cajuns at Antietam in a formerly-all-Virginian regiment without issue.

Randy DeCuir's book, "Arcadian Guards: The French Speaking Confederates of Company F of Mouton's 18th Louisiana Infantry: & Company I of the Consolidated 18th Yellow Jacket Battalion" is one of the few books that focus on predominantly francophone units, and there are websites that detail the integration of originally Arcadian or white Creole units within the wider Confederate Army. But by the 1860s, few communities were solely francophone in Louisiana. Most people had learned some measure of English, especially if they were the sort to fight for the CSA instead of vanishing into the backwoods when the draft hit hard. Or, as the all Creoles of Colour unit did, they switched allegiance to the Union Army, where polyglot units were, from all appearances, handled without much issue. Early in the war, the CSA had units "elect" officers from within their ranks, so likely, some French-dominant LA units had French-speaking commanders, but after 3 or 4 years at war, that wasn't an option and more than likely, any Confederate soldier who had been French-only prior to the war had learned enough English from encampment and units being combined to function perfectly well in the army, especially if they had been in for a while.

This is admittedly an overview of what's been written and documented about francophone units. If anyone has it as a specialty and knows more, I happily defer.