I think the OP is mixing the two. Ayway, this scenario only makes sense if it refers to Nap III, so here we go.
If you look at
the map, the French pretty much conquered Mexico in OTL. The problem? It was a pain in the ass to held it. The things Europeans and the French themselves wrote about this war at the time look a lot like how the Vietnam War was seen a century later. And when the ACW ended and an US intervention became a real possibility, it was obvious that France wasn't in a position to continue this doomed crusade and couple it with a full-scale war against the USA.
There are quite some points that might benefy a lot the French, though (not necessarily needed to happen in the same TL):
- Replace Bazaine with another guy. Not bad at the battlefield, but he was an ass out of it. He openly ignored Maximilian and described himself as Mexico's Bernadotte, implying that he would be ever the top head there. When he was recalled to France, he ordered the destruction of all weapons that couldn't get on the French ships, depriving the Mexican imperialists of them just for the sake of it...
- Find another guy for the throne. Maximilian was a liberal fighting liberals and supported by conservatives. Seriously, that wasn't going to work ever. the throne was actually offered to other men that refused:
Enrique de Borbón, brother in law of Queen Isabel of Spain was one of them, IIRC. What if him or any other had accepted? Could an actual Spanish-speaing emperor of the last real royal house that ruled Mexico, be better received by more Mexicans?
- Better yet, why not a Mexican? The man planned once to be
Agustín II of Mexico was still alive at the time, and had commanded Mexican troops during the Mexican-American War. He was unmarried, but he had nephews (one of them became Maximilian's adopted heir). He really could have been popular and would erase the complaints of "un-Mexican-ess" of the regime from other countries. This looks so dam obvious that Napoleon III strikes me as a complete idiot for passing it over.
- Retain the early Spanish support. Isabel II had toyed for a time with the idea of installing a sister Bourbon monarchy in Mexico. That could have been achieved both with the Enrique option or by rranging the marriage of the heir with one of her infant daughters. In return, Napoleon III receives more troops, more cash and the possibility of using Cuban bases and the idea that the whole matter is a single French adventure driven by greed which in turs condemns France into political isolation is taken away.
- Retain the early British support. This is almost impossible and probably needs a Trent War scenario to work (i.e. France helps Britain in Angloamerica, Britain in turn helps France in Mexico).
- Screw really badly the USA for the next 20 years (i.e. Trent War and/or Confederate victory) while the Empire consolidates.