Interesting notion and perhaps not quite as unlikely as some would suppose.
The Mexican intervention arose from Mexico’s failure to repay loans. The usual procedure was for the lender to then force repayment by force of arms and both France and Britain send naval and ground forces to Mexico. They captured several ports and blockaded others. At that point, Napoleon decided upon a major intervention. Mexican Catholics (the upper crust and church) had asked for his support against the Mexican liberals who were taking away their privileges. The French Catholics supported their request and the offer to accept an European Catholic prince was made. Napoleon needed the support of the French Catholics and there was a popular belief that Mexico was rich in resources.
Britain did not go along with this and left France to go it along, with Austria which provided some troops and a Habsburg prince and even Belgium.
Historically, Britain did worry about American intentions about Canada. It would make sense strategically, if the USA was weakened and/or a powerful friendly nation (either Mexico or the CSA) arose in the south so the USA would be “encircled”. It is not inconceivable that Britain would have quietly assented to a French campaign to support Mexico and the CSA under its much loved divide and rule strategy. France would be committing resources and would not be able to either threaten British colonial interests nor could it engage in European adventures, thereby maintaining the balance of power. If they succeeded, the USA would be less of an (economic) rival and if it failed, France would have been weakened and become more of a client state.
So how does France and the USA/CSA compare militarily in the early 1860’s? Simply put, France is superior in every way. The French navy was competing with the British to which end it scored several technological coups. It didn’t quite achieve its goal of matching the British but it was easily the second most powerful navy in the world. As for its army, the French Imperial Army was rightly considered the best in the world, notwithstanding its later defeat in 1870 which was down to its leadership and tactics.
In the 1860’s it was an experienced regular force with state of the art equipment. Its performance in the Crimea and to a lesser degree in Italy had won international plaudits. Compared to the British army, the French was more professional and more able and simply put the British to shame in the Crimea. The USA/CSA forces were not in the same league as the French. While the Americans would become quite accomplished after several years of campaigning, they started out as armed mobs with little formal knowledge of military art & science and could not have prevailed against the French in the type of battles fought in the ACW.
Therefore, the French would have the means to materially influence the ACW. It deployed 400,000 men in the Crimean War (dwarfing the British contribution) and deployed 250,000 men in Italy in 1859. Even the relatively minor Mexican expedition was supported by about 40,000 men while the ridiculous pastry war saw a French invasion with 30,000 men. So the French could deploy large numbers of troops and maintain them overseas.
How and to what end would have been a matter of strategy. Napoleon’s track record isn’t particularly good in that regard but he did have some competent Marshals and ministers who could have provided the necessary advice.
