If Napoleon III decided to do so, he would have been going at it alone. He would not have done so without the British or any other great power for that matter.
I think that
AshleeNova is premising her thread on the idea of "What If?", regardless of the OTL problems of doing so. IOW, Imperial France supports the CSA, so "now what?" The obvious barriers to a solo French intervention in the American Civil War are openly manifest, and don't need to be gone into here. There have already been countless threads on how it couldn't and wouldn't be done.
Despite the British Aristocracy that sympathized with the Southern Planter class that dominated the Confederate government and society, the British had been the leading advocates of abolishing slavery since the early 1800's and the government would not have risked political fall out for supporting a cause that was fighting to support the institution of slavery. Even at the highest level, Queen Victoria's influential husband, Prince Albert was an avowed abolitionist.
All true.
However, on AH.com there is the persistent phenomenon of what are called the "High Tories". (1a) (1b) People who blame the USA for the fall of the British Empire (2) and are frustrated that the lack of Foreign Intervention in the ACW stymied the only real chance that ever existed for breaking up a continental-spanning constitutional republic in North America. (3) The idea being that a divided North America would have prevented any American meddling in European affairs, and butterflied away the anti-colonialism of Franklin Roosevelt. (4)
1a) Basically, those who dream of the good old days when the British Empire benevolently (5) ruled one quarter of the world's surface and could dictate policy to any nation that wanted to do business on the high seas, since the Royal Navy back then was more powerful than every other nation's navy in the world combined and then some. Not to mention the British monopoly of nitrate manufacture, that wasn't broken until 1913. And in 1914...
1b) Otherwise known as Rule Britannia Forever types or Sun Never Sets Britons. American Exceptionalism with fish and chips. But since the British Empire no longer exists, and you actually have right wing politicians in the US bragging about their championing American Exceptionalism, you don't see High Tories drawing the kind of censure that the US version does.
2) And prefer to give a pass to WWI, the Central Powers, crushing WWI war debts, failure to reform Imperial rule post-WWI, Amritsar, interwar disarmament, the Great Depression, Appeasement, WWII, the Axis Powers, Japan's destroying the myth of the supremacy of the White Man, crushing WWII debts, the crushing Socialist victory in 1945 that resulted in 8 straight years of Labour rule, Suez, shall I go on?
Far better to blame the Evil Empire that is the USA and their dark overlord, Franklin Delano Roosevelt.


Essentially, your average High Tory sees the US Civil War as being like the Revolutions of 1848: A great turning point in history where history failed to turn. In 1848, history turned in favor of the aristocracies. In 1861-65, history turned against them. The Union "wasn't supposed" to win the ACW. Or at least its a damn shame that they failed to lose.
3) And what happens in the 20th century to Europe's democracies without a powerful continental democracy in North America to help back them up when things on the European Continent turned to poo? You can't even say handwaved. When discussing the ACW, High Tories ignore the natural consequences to be faced over the next 150 years. All you get are comments about "Super Commonwealths" and "Soft Landings" for the Empire. But no Great Wars. Pax Britannica Forever.
4) FDR was willing to fight to save Great Britain (and France and the rest of Europe). But he wasn't willing to do so for their empires. Not after the Imperial powers did all that they could to restore the
status quo antebellum plus swallow up the German colonies AND the Ottoman Empire outside of Turkey itself. I've seen posters state emphatically that it very much WAS the duty of the US to restore the empires of their allies, by both blood and treasure, and state so without any apparent sign of self-consciousness.
5) If by "benevolently", you mean better than any other colonialist power. Which isn't saying much.
Recent growth of the cotton market in India made Southern cotton not as vital to the British textile industry and despite tensions early in the war between the British and the Union government, continued beneficial trade between the two countries made it recognizing the South not in their best interests.
POTCS Jefferson Gump's

decision to burn all that cotton on the wharves to "starve Europe" of its cotton at a time when the CSA was itself starved for cash and the Union blockade scarcely started yet turned out to be an excellent way to force the British to massively expand on cotton development in Egypt and India. However, the grain of those cottons were poorer than the CSA's, and were problematical at best at that time in their use in textile mills.
That said, Napoleon III's recent friendly relations with the British and reliance on that friendship would have prevented him from recognizing the South.
Again, the OP is waving that for purposes of discussion.
As for his venture in Mexico, he no doubt counted on the fact that the Union of whom won big victories at Gettysburg and Vicksburg in July of 1863, were destined to win the war.

It was the very timing of his invasion, 1861, that made clear Nappy III was counting on a
Confederate, not Union victory. Otherwise, you'd see what exactly did happen OTL: Once the last Confederate troops in Texas under Kirby Smith surrendered, it was Game Over for Nappy's Mexican adventure. With an 1865 standing Union Army of 500,000 plus troops, resistance was futile.
Even with victory, he may have assumed no "quick" end to the war was at hand and even the eventual Union victory would have resulted in many years of recovery and the US would not have the stomach for an eventual fight on behalf of Mexico.
Actually, with surplus Union Army war supplies and arms up the yin-yang pouring across the Rio Grande to the Mexican republican rebels, our valiant Mexican neighbors hardly needed American help in the matter of additional troops. The Mexicans were fighting in their own country and for the first time fully equipped and supplied against an enemy operating between 3000 to 5000 miles from home, with no Panama Canal to skirt around all of Mexico's vast coastlines.
Great powers do not stay great powers by recognizing lost causes that were unpopular because they were fighting to preserve the institution of slavery. Napoleon III like his British counterparts understood this.

Too bad he didn't show this caution 5 years later...