I don't know that this would have been that big a hurdle for Cuba. Louisiana gained statehood despite being (at the time) mostly French-speaking and Catholic, and had a large black/racially mixed population.
Nearly all the blacks in 1812 Louisiana were slaves. Louisiana's free
gens de couleur were something of a legal anomaly, but their status was second class even then, and was gradually reduced toward the equivalent of "free colored" in the rest of the U.S. So Louisiana did not pose a threat to white supremacy.
The Cuban population included a much larger number of rich or politically active mulattos and blacks, including military officers and upper-rank LEOs. And in the U.S., there was a black U.S. Representative from North Carolina (George White, the last black Republican from the South until Senator Tim Scott today). A few years later, black Republican Oscar DePriest was a Commissioner of Cook County IL (where Chicago is). So even in the U.S. proper the "color bar" was not solid.
Thus it would not have been practical for mainland Americans to export the color bar to Cuba, and that would mean "black" U.S. Representatives and Senators. Therefore, IIRC, Southerners were opposed to any annexation of Cuba.