If it's quite prevalent in literature, one has to examine why it became a meme in the first place. In many ways, there are a number of things that come up in CSA wins timelines.
1) Confederate Cuba: It's a fun thought, but the politics of buying Cuba just don't match reality and the OTL Spanish-American War was the result of a freak accident that doesn't likely get replicated in the new timeline. Any Hispano-Confederate War would have to break out for a completely different reason.
2) Secessionist Mormons: Things would have to get pretty bad for this to actually happen. The Federal government would have to agitate them enough to make it happen. Either that or the British are extremely successful in penetrating the American West (unlikely) in a hypothetical Trent Affair War.
3) The CSA doesn't do very well and splinters: There's no reason why they couldn't do well. A successful nation in the 19th century didn't necessarily have to be a great power. Nations still traded with the United States and Brazil even though they had slavery.
4) Balkanized USA: This isn't very common, but there just isn't enough support for balkanization in the mid-19th century. In fact, the absence of the CSA would kill much of the sectionalism in the country. Maybe the UK decides it wants a puppet California in the event of the Trent Affair War, but by and large, the thought that the secession of the CSA would have been a threat to the very fabric of the Union was fear mongering. An independent CSA did not mean there would be an independent New England, an independent Utah, an independent California, or whatever.
5) Amendment banning secession: While it's not far fetched to think the Constitution would be amended to prevent secession, there's no reason to think that amendment would absolutely be ratified either. It's actually very hard to amend the Constitution and the threat of secession would still be a useful political tool especially in the wake of the ACW.
6) Slavery is carried on in perpetuity: While slavery was profitable when many nations abolished it, the late Robert Perkins often pointed out that real world factors would ultimately lead to the demise of slavery. The boll weevil is a particular menace that would drive many plantations out of business, which would ultimately cut down the demand for slaves and drive down the prices. That would ultimately help kill any under the table slave importation because the risk wouldn't justify the reward. You could work out the dynamics of immigration issues on your own, but the clock was ticking against slavery. While they couldn't stop it in Richmond, state governments certainly could if the influence changed drastically enough to force gradual (most likely) or full emancipation (extreme circumstance) in a state. It wouldn't improve race relations, so Jim Crow would probably prevail simply to prevent job competition with freemen (and due to racism of course).
7) Texas secession from the CSA: This is actually quite ridiculous. Texas barely had the majority vote to approve secession from the USA. Rather than remain independent, they joined the CSA.
8) The USA and the UK continue as mortal enemies: This is actually no guarantee either. There's actually zero benefit to either nation. They would resume trade and the UK would likely do whatever it could to play nice because defending Canada against a very hostile USA is a military nightmare and plus let's not even get into the economics of having to keep a strong standing force in Canada. It hasn't stopped nations before, but saner heads prevailed quite a few times in the 19th century.
9) The USA and CSA continue as mortal enemies: Like with the UK, there's no guarantee that this would be an issue. Certainly there would be revanchists within the USA, but must they always prevail? Maybe they actually could get along.
10) Nobody really learns anything (or no major butterflies): Depending on the war, the nations of Europe might pay more attention to military lessons from the war. If the UK and France get involved in a Trent Affair War scenario, then you can bet other European nations are going to be paying attention and making notes of what military doctrines work and which ones don't. Now whether the lesson is learned is a different matter entirely, but this all depends on the sort of war that unfolds.
Those are just ten things I can think of. Feel free to dispute points, but it plays into the general theme of Mormons getting screwed in CSA wins timelines.