Having reviewed this thread - I suggest a PoD: the Gadsden Purchase falls through. The Purchase was made to secure a passable southern route for a Pacific railroad. If it fell through, no southern route would be possible.
The knock-on from this: one reason for Douglas putting forward the Kansas-Nebraska Act was in exchange for southern support for the middle route for a Pacific railroad. (Allegedly, he wanted this route because it would link to Chicago, where he had real estate interests that would benefit,)
If there is no competition from a southern route, Douglas has no need to do that, so he doesn't. No Kansas-Nebraska Act.
This is an interesting POD, since taking Kansas-Nebraska out of the equation in the first place allows for more deviation from OTL's reigning attitudes. That is: if slavery in Kansas is never a real option, the Compromise Line is never really questioned, and Southern attention is more on Caribbean gains... then the combination of there being no perspective for slavery in the West and there contrarily being a success of that 'Caribbean strategy' might greatly reduce tension over the Western states. This would probably work to delay any Civil War even longer.
Southerners, acknowledging the futility of trying to break the Missouri Compromise and plant slavery in Kansas, focus instead on Cuba. The Black Warrior affair leads to war in 1855-1857, when Spain concedes Cuba and Puerto Rico to the US. (ISTM that the US would need to occupy Puerto Rico for military reasons, and would want to annex it along with Cuba,)
In 1859, East Cuba, West Cuba, and Kansas are admitted as states. (Minnesota was admitted in 1858, so the free/slave balance is retained, except for California and Oregon, which were for some reason exempted.)
The Whig Party breaks up as OTL, in part because many Southern Whigs (but no Northern Whigs) support the war with Spain. The Republican Party forms in 1855. The American ("Know-Nothing") Party also forms, opposing the war aim of annexing Catholic Cuba.
Pierce wins easy re-election in 1856, on the strength of the victorious war (not quite over, but the US has taken Havana and Santiago, and destroyed a Spanish squadron off Florida).
Cuban statehood brings out some bitter feelings, but anti-Catholic sentiment simply is not strong enough. The Democrats remain united, with the Slave Power sated by Cuba. Douglas wins the 1860 election against a Republican and a weak Know-Nothing.
Puerto Rico and Nebraska are admitted to statehood.
At some point the slavery issue is going to boil over. OTL, it boiled over because the South pushed it too hard. ITTL, not so. Pro-slavery expansion in the Caribbean would offend many Northerners, but not as critically as slavery expansion into the Territories, and certain not as much the flagrant and brutal effort to force slavery on Kansas.
This seems like a realistic sequence of events. The US getting Puerto Rico as well was something I hadn't considered, but which does seem like a realistic outcome. Regarding the slavery issue: see below.
Some questions arise:
Would Spain fight for Cuba, or be cowed into selling?
Would the US become more interested than in OTL in acquiring the Dominican Republic? (It seems probable that the *Spanish-American War would pre-empt OTL's reconquest of the DR by Spain.)
Spain would fight, but would start losing pretty fast. At that point, it might consider ending the war with 'voluntary' a sale of Cuba and Puerto Rico. Indeed, this outcome would galvanise the Southern idea that the Caribbean ought to be theirs (the idea of a slavery-based 'Golden Circle'). As you mentioned, lots of Northerners would be unhappy about adding Caribbean slave states, so expect no formal US action. On the other hand, lots of Southern veterans of the war agains Spain can be tempted into joining a filibuster. Expect it to go like that.
An idea: the POD you suggest (no Gadsden purchase) has the effect of derailing OTL's filibuster to capture Sonora. This changes the life of the man who led that filibuster-- AH favorite William Walker. He still goes filibustering about, but doesn't meet his end as in OTL. Instead, he's the leading advocate in the South for annexing Caribbean islands, and eventually leads the filibuster that captures the Dominican Republic.
In any case, the conquered, now slavocrat-ruled Dominican Republic would petition for annexation. Presumably, the North could use this to get the South to agree to the vast areas of the West that are north of the Compromise Line being cut into a lot of new states/territories. Per OTL, the North can get about nine more free states out of the West, after Nebraska is admitted. On the other hand, the South can only expect three more slave states in the West (Indian Territory/Oklahoma, New Mexico, Arizona).
Eventually, you end up with 29 free states (or 31 if Alaska and Hawaii are added on schedule) and 21 slave states. Either way, that won't get any amendment banning slavery ratified. The South feels more confident, and you can end up with a stalemate. (An additional note: we may also see further Southern-led filibusters, which could add such areas as Haiti, Nicagragua or Panama as slave states-- presumably as a way to 'balance' the annexation of Alaska and Hawaii.)
Could such a stalemate avoid the Civil War altogether? Perhaps. A POD that casts the Southern gaze to the Caribbean from an earlier point, while making them 'give up' on the West, would remove the big clashing point of OTL. Some cycical Northerners may even tentatively support Southern filibusters in the Caribbean, thinking "Well, this gets those lunatics out of our hair, at least! Let's go settle the West with
our people!"
When the US acquired Louisiana, it had a significant population of free gens de couleur who had substantial political rights. With mass migration of Americans to the area, these people were gradually stripped of their rights and elevated status; reduced to the same status as "free colored" in other states. Cuba would have a much larger and more important mixed-race population segment. IMO, there would not be sufficient American in-migration to force the sort of social change that happened in Louisiana. It's quite possible that some of those men would be elected to Congress. How would Southerners react to that (bearing in mind that these men would be pro-slavery Democrats)?
Would the example of Cuba undermine the strict color bar enforced in the US? Perhaps even leading to acceptance or at least tolerance of interracial marriage?
This would be nice - a positive side effect of a horrible regime - but I think you can forget about it. There are no Fourteenth and Fifteenth Amendments here. Expect ruthless discrimination, and a situation where only a small sliver of the insular populations is allowed to vote. Which sliver? The whitest sliver, of course. Catholics who are white enough and who own enough slaves and who say all the right things ("States' rights! States' rights!") will likely be tolerated, some as they were eventually tolerated in the North-East. But recall that even in the second half of the 20th century in OTL, Kennedy faced attacks based on his Catholicism. (He had to neutralise the issue, telling a group of Protestant ministers: "I am not the Catholic candidate for president. I am the Democratic party’s candidate for president who also happens to be a Catholic. I do not speak for my Church on public matters – and the Church does not speak for me.")
These insular states are going to be minority-led: basically plantations write large.
In a possible Civil War, would the island states join the Confederacy? IMO, they would. Would the Union move to occupy them? It would seem they must.
If there is a Civil War, the ruling elite will definitely join the CSA. The vast majority of the population, of course, will not be part of that ruling elite. If the USA had gained only Cuba, I could see Southerners being relatively more numberous. If white migration to the islands is spread across several, however, the white elite will be very small. Local pro-Union (or at least anti-Confederate) uprisings would be near-certain.
If the Union wins this Civil War, would East and West Cuba become Republican states with black and mixed-race voters in a majority? Or would Redeemer filibusters from the mainland enforce white supremacy? How wold a breach in the color line affect American society?
They would be Republican, provided the Union doesn't just spin them off into independent client states. There would be no breach in the colour line, the North was also still very racist, and for that exact reason they might decide to leave those islands out of the postbellum USA. They'd end up sort of like Puerto Rico in OTL, I'd expect. That's less than ideal, but I do think it would probably go that way.
All this, of course, assumes that there is a Civil War. In this specific scenario, that may not be the case. As I said: an extended stalemate might occur instead. How that would eventually end is unclear. North and South would increasingly drift apart, but with the West all hashed out, war becomes less and less likely. The North can't abolish slavery, the South
can preserve the status quo. I see two outcomes to that. One is that the USA splits apart peacefully. That would mean that either the North secedes, or both sides decide to each go their own way. This is not likely to be done lightly, but the second option would be that the South 'holds the North hostage', eventually demanding vast economic concessions in return for abolishing slavery (along the lines of 'we want to be fully compensated for out losses if we are to abolish slavery'). I don't think the North would accept that kind of demand. At that point, option one starts becoming more realistic...