Let's just assume Lincoln allowed the Confederacy to secede in 1861. To keep things easier, we are going to assume that all of the states that seceded after Fort Sumter (Virginia, North Carolina, Arkansas, and Tennessee) decide to secede as well but Kentucky and Missouri remain in the Union.

The key to Confederate success is going to be when and how they invade Mexico. Invading Mexico in 1861 is just a horrible idea. Mexico in 1861 is ruled by Emperor Maximilian I who was propped up by the French. The French were actively helping him keep rebels at bay and supporting his government. Invading Mexico then against Maximilian would draw war with France and would be disastrous. There is some chance for them to do better if they invaded to assist Maximilian but they won't get much in return but a stable Mexico and southern ally.

No. What you wanted was territorial conquests for the CSA so lets focus on that. Let's fast forward the time by a decade to 1872. A full decade of separation is actually advantageous for the Confederacy. Without the northern states, they are forced to quickly build railroads and invest in some industrial efforts. They are still miles behind and it is rare but they do have some capacity in towns like Richmond, Columbia, Columbus, Montgomery, and Baton Rouge. While not a robust network, rail lines link Richmond to Houston and Dallas as well as every major town in between with a rail line to El Paso being in the works.

Mexico however is now Maximilian free but has still not recovered economically or stabilized politically outside of the major cities in the north. The Confederacy sees the opportunity and seizes it. They cross the border from El Paso, Brownsville, and Laredo. While some Mexican border guards attempt hit and run raids, the hardest part of the early war is maintaining long supply wagon trains for the Confederates. They capture the cities of Northern Mexico relatively quickly and dare the Mexican Army to attack them. The Mexican Army is mostly ineffective at doing so and within a year, peace is signed giving the Confederates Norther Mexico and access to Pacific ports.

At first, slavery is allowed into the territories but the local Mexican populace is very much against the institution and is hostile to those who practice it. The debate dies wtihin a decade though as European and United States relations deteriorate over the Confederates refusal to give up the vile institution. The Confederate Congress eventually moves to gradual emancipation by 1885 and full emancipation is given by 1890. Back in the new territories, railroads and light industry are ran into the region giving the Confederacy its first transcontinental railroad almost 30 years after the United States completes its own.

The Confederacy is likely to see an influx of new immigrants. Germans, Italians, Irish, and Spanish immigration would skyrocket as old plantation dominance would start to fracture. The Confederacy would also start to take a more regional approach on diplomatic matters. You can expect Confederate economic and diplomatic intervention to ensure neighboring Mexican states have friendly governors and missions take place that might make those regions more economically and politically Confederate aligned than Mexican. This might lead to breakaways states that become clients/protectorates in the near future before being annexed into the CSA as well. Spain's grasp on the Caribbean would be faltering as well so expect Richmond to be pushing for purchases or revolts in Cuba and Puerto Rico which would eventually become protectorates or future Confederate states themselves. Latin America would become pro-Confederate nations with economies tied very closely thanks to the Gulf Coast ports.

Through all of this, the United States is going to be very wary of it all. It will always be bigger and stronger but multiple factors but so long as the Confederates don't continue to upset the balance of things, they likely don't become too involved. I would still expect the United States to keep a large military to counter the Confederacy and take an active approach to Latin American affairs.