*1865: Sterling Price is able to escape into Mexico and takes a number of CSA troops with him. There is also some emigration from the CSA where wealthy planters seek refuge from a government they believe hostile to them. Robert E Lee unfortunately passes away in March 1865, leaving command of the Army of Northern Virginia to J.E.B. Stuart. A Confederate underground rises away from the major cities that reaches even into the West, where its most notorious members (OTL James Brothers) succeed in harassing the Transcontinental Railroad and robbing banks across the Midwest for years. Meanwhile in Mexico, Price is able lead forces into northwestern Mexico and smash what resistance remains.
Price was mediocre as a commander in OTL. There’s a couple points during the Overland Campaign where Lee’s behavior was nigh-suicidal, so his death in 1865 is very credible. Stuart was not the senior commander in the AoNV and by 1865 the main thing holding that army together was Lee. His death hastens the collapse of that army and the Confederacy.
Large numbers Confederates fled to Mexico in OTL. The James brothers robbed banks and railroads for years in OTL. The first Klan and similar groups took a decade to forcibly suppress in OTL. Why would all of these be more successful in TTL?
*1866-1870: As the US recuperates from the guerilla activity and the war dies down, Maximillian builds a surprisingly stable government out of Confederate resources, European investments, and local ingenuity.
What Confederate resources? And it’s going to take a long time to make Mexico stable – the liberals hated him because he’s a foreigner, and Emperor, and propped up by European power. A lot of the Conservatives hated Maximilian for being a foreigner and a liberal. If he suppresses the Juaristas, there’s a very strong chance Maximilian will be overthrown by a Conservative coup.
He is opposed by some of the conservative nobility and the Church but they are placated by his raising of his children as ardent Catholics along with his fathering two children (a son is actually that of Sterling Price from an affair shortly before his death in 1868 while a daughter is actually his).
Points to you for including something that doesn’t further you main thrust of wanking Mexico. OTOH, this isn’t very likely. Sterling Price was not Earl Van Dorn and old enough to be Carlotta's grandfather. Maximilian having children does not make his rule more stable. He and Carlotta had already adopted a couple grandchildren of Iturbide and named them as heirs. There will be plots and possibly coups or a revolt in favor of Maximilian’s adopted sons.
Guatemalans rebel from their old rule early in 1870 and by plebiscite seek to become part of Mexico because of the prosperity and increased freedom. Secretly many hope that the Empire can reunite Central America and break the backs of the cabals of large landowners that rule the banana republics. El Salvador follows when the price of indigo falls and border skirmishes with Honduras lead to an invasion of that nation in 1873. Washington protests but its navy is in a state of disrepair and the pains of reconstruction are still readily apparent, this causes Maximillian to quietly embark on a naval program of his own with more power concentrated in the Pacific. The Mosquito Coast (of Nicaragua OTL) is annexed by a plebiscite there in 1878 and the first steel steam cruisers of the Mexican Empire take to the sea quietly about the same time.
Maximilian was a naval enthusiast, but I’d expect him to focus on the Caribbean, not the Pacific. Even in disrepair, the US already has a navy and they’ll look on the rise of the Mexican navy with the same enthusiasm the British showed towards the rise of the German Navy. There will be a naval arms race, which the US will win. For that matter, Britain has interests in the area and they won’t like Mexican expansion, either.
Very few smaller countries have willingly giving up their sovereignty to join a larger nation. The attempts for Central America in OTL involved uniting to protect against Mexico and they always fell apart because none of the nations involved were really willing to give up their sovereignty.
It’s probably going to take Maximilian at least a decade to get firm control of Mexico. Starting circa 1875 Maximilian should be able to start conquering Central America. If this doesn’t trigger a response from the USA or Britain or result in Central America uniting against Mexico, he might finish the conquest by 1880, but it will probably take longer.
*1880-1890: Serious discussions begin in Mexico City about building a canal across the isthmus at Nicaragua lead to increased trade and a direct rail link between Managua and Mexico City. Maximillian coerces British, French, and German investors into believeing that the canal system will work and that it will benefit all their nations. Mexico and Nicaragua will administer the canal while the Europeans will be given tolls on it until 500% of their investment is reached. Mexican ships are also declared exempt from the toll, causing several ships from less powerful nations to register with Mexico and pay fees into His Majesty's coffers.
You can’t coerce foreign investment. And Ferdinand de Lesseps, builder of the Suez Canal, has already been fundraising for a Panama Canal, so I'd expect more investment there. Mexico's only going to get enough foreign investment after the French attempt fails, and be in competition with the US. Mexico a Nicaraguan Canal won't start till the early to mid-1890s and will take a decade to complete.
He rules over an Empire that profits greatly from the Canal and becomes the impetus for US naval rearmament as the Mexican navy outguns that of the US by 2:1 in 1886.
This is flatly impossible. The US has vastly more industry and population as well as an established navy. Mexico cannot win a naval arms race. They cannot match the US gun for gun, let alone double the US total.
Pancho Villa becomes a dictator in the north and eventually invades the Empire, Maximillian dies a martyr's death while visiting an aid station when raiders kill him at the age of 83 (he would have died of cancer within the year) allowing the Iturbide family to return to power.
That assuming neither of Maximilian’s adopted Iturbe sons attempts a coup before his death, Maximilian passes over his natural son for them, and his natural son is okay with this. You’re more likely to have a succession war than a peaceful transfer of power.
Mexico also has a strong history of trade with Japan, perhaps they play a key role in opening up the new country and begin a profitable relationship between two Empires.
Japan was opened up before the POD.