At first, I thought this would be a no-brainer.
After all, in our timeline, Mexico hadn't produced much of a showing. Essentially, they'd gotten their butts kicked badly in the Texas Revolt. Then they'd gotten their butts kicked badly in the Mexican American war.
In both these wars, they'd only been able to field relatively small armies, the armies were poorly equipped, logistically inferior, and badly lead.
In our timeline, there's no real sign of that changing. They had a pretty good guerilla war in the Civil War period, and periods of internal strife up to the Mexican Civil War.
In the meantime, the Confederacy in the Civil War was able to field a sophisticated first rate army, well equipped, well lead, in huge numbers.
So it seems to me that it would amount to a big swat.
Likely it still is.
I don't know that there would be sufficient butterflies to really change the equations. But there might be a variable or two.
A Confederacy might be substantially less effective projecting power. They fought the civil war on home ground. Going into Mexico might be a challenge to logistical capabilities, who knows.
And if their Mexican incursion ends up going against France, who knows. The thing with this is that the Confederacy was pretty tolerant of Maximillion, and might continue to be. On the other hand, alliances like that change on a dime. So the CSA might want the French out, might or might not force the French out. Might occupy Mexico themselves. A guerilla war is always brutal on an army.
But supposing the French don't leave and it turns into a stand up fight.... Who knows, it might be the thing that reconciles the French and Mexicans. At the very least, the French will be a lot better at fighting a stand up war than a guerilla occupation.
We might have a butterfly in this timeline where Mexico has a much better time getting its act together and becomes a more robust and viable state that could pose a real challenge to a CSA. Who knows. Off the top of my head, we might see a situation where the USA builds up Mexico in order to balance the Confederates.