Without a timeline to critique all of this boils down to navel gazing.
Let's say for a moment that I were to write a timeline in which the CSA invades Mexico (I won't). In this world Bismarck might say, "A special Providence takes care of fools, drunkards, and the Confederate States."
While I would try to keep things logical (people wouldn't automatically do things they are not prone to IRL), the CSA might have caught some amazing breaks and the US might have been considered cursed by the gods. For instance, it would have been Grant dying at Shiloh instead of Johnston. The mental issues that we now understand Sherman to have suffered from would have overcome him, rendering him unfit for command. And despite how cliched it has become (Thanks again, Turtledove), that pesky order 191 doesn't go missing and McClellan's caution costs him the battle... or something along those lines. Lincoln would have kept trying to find a general who could defeat Lee, and heck, just for gits and shiggles, a bullet from a sentry narrowly misses Stonewall Jackson at Chancellorsville. The above are all vagaries of war and could have broke either way.
My point is that Alt history is what the one writing makes of it. Sure, the POD (apart from an ISOT or some other ASB handwaivium) needs to be plausible. The thing about butterflies that we tend to forget is that subsequent events after the POD only need to make sense given the previous divergence. It really doesn't have to make any sort of sense to the original timeline IF it coherently follows on the previous divergence.
If I were writing a TL of CSA invasion, I would find a way to neuter outside intervention, unless the goal of the TL was outside intervention. If I were writing this TL, I would also include a POV that showed the horrors of slavery, probably through the lens of a slave revolt. The point is to show the reader that all isn't well in the story and that while the South may have won their first big fight, there's a larger one on the horizon.
Honestly, if I were writing any CSA victory timeline, there are several things I would want to focus on, first would be to explore more fully Southern views on the role of government in civic life. Many positions the south took at the state and local level tended toward a statist view of things. Sure, they were rabidly anti-federalist, but that's not the same thing as being anti-statist. I'd also explore the tension between those who understood the need to industrialize with those who didn't. Readers would be surprised to learn that there were plenty of folks in the south who understood the dangers of an economy built upon a single export. And lastly, I'd explore how slavery devalued labor and cheapened humanity.
It depends how you see Alt History. If you see it as a fund thought project where you can wank a country or dump it, sure. I'd agree with you. But that's all it is, not an actual possibility and not an actual answer to a "what if" question. Don't get me wrong, those have their merit. For me the value of Alternate Histories is to examine the reasons why things are the way they are. It helps us learn details about history. I see something and I wonder what would happen if a person does something differently so I read about who these people were, what they did, why they did it, and other pertinent facts. It's a fun way to learn about history. A TL can have a sort of art to it as you suggest, but that is what it is. An Art, a story to entertain but still provide some educational value and food for thought. There is a limit to this, because if you go too far, then that means we'd be making things up like say, a CSA population of 40 Million in 1900, or propagating false information which has no value at all.
I've read/am reading a few timelines here and there that are an art and I enjoy them. For the most part they don't take me out of it by being ridiculous, they haven't been ridiculous. But making a successful (forgive me) sea lion operation without any serious changes to Nazism, would be too much to ask.
The OP asked if it is possible, and since this isn't a timeline, we can't conjure up an "artistic" TL to answer in the affirmative. We have to stick with the most possible scenarios and not manipulate the odds and make the CSA win the lottery every time. Because we can play the same game with Mexico and use the CSA victory as a POD/Butterfly that changes Mexico's luck (It has had the worst of luck).
After Juarez the very real threat of CSA invasion begins to sink in and Diaz begins to work closely with the US to build up his industry, and focus on creating a navy (Politics prevented the construction of a defensive navy in the OTL). Maybe even a navy similar to that of Brazil's by 1900. Sure this would strain the government's budget, but some friendly loans from anti-slavery USA and UK would go a long way. I could maybe sway things to say that the UK is so worried that Brazil might ally with the CSA to preserve both country's slavery, that it sees Mexico as a wonderful partner to keep both nations in check. It might spur friendly relations with the US to create a great Anglo-American alliance. Maybe Diaz is so focused on the threat that he agrees to be conciliatory to Madero and prevents the Mexican revolution all together and retires in say...1914 and Madero wins the subsequent elections and begins the democratization of Mexico. The UK and the US help Madero to quickly put down a Huerta coup allowing Madero to rewrite the constitution by 1917 establishing a firm democracy yet avoiding a one party rule. The stability allows increased trade and immigration to Mexico. The new constitution outlines liberal initiatives like education which raise Mexico's literacy. Mexico even sends some symbolic units to fight in WWI, maybe a regiment or two attached to US forces....but this does not answer the OP's question at all, because I
made things happen.
But doesn't that answer the OP question? That the CSA would have to be very lucky to be able to do it. It is plausible, but only if it is very lucky and if the butterflies don't make Mexico better due to the rise of a new threat and the US's reactions to the new order.
I guess the take-away is that the further away you get from the point of a CSA victory to an invasion of Mexico the greater the uncertainty about things. No surprise, as every year should introduce more and new butterflies.
I would agree with this. A 1867 Mexico would be the easiest Mexico to invade until 1911. If the CSA won and was expansionist, I doubt that Diaz would just sit with his arms crossed. The USA would definitely be willing to arm Mexico to the teeth and openly trade with them. If Diaz asked the US for warships, the US would only ask how much they could afford. Under Diaz's "presidency" Mexico experienced stability and economic growth and modest industrialization. If the US was helping out Diaz, it could have better growth. The problem is that the longer the CSA waits, the more prepared Mexico and the USA become.
Not to mention, the CSA would need a Casus Belli. What would it possibly be? They have land, we want it? I guess they can just simply declare war...good luck being trusted by other countries because it would now be super easy for any nation to justify material aid to Mexico....like say the USA.