I've always looked at FP as an alternate reality; what the Star Trek universe would be like, if the South had won the civil war.
Isn't it much simpler to look at it as a separate creation that was made in the 1950s and reflects the assumptions of that era? Let FP be its own entity. It did come first, after all.
Besides, even if the South had won the Civil War, that doesn't mean there would never have been racial equality. First off, the Confederacy was trying to leave the USA, not conquer it. Even if they'd won the war, it would just mean that the US would be smaller today, not that it would've been taken over by slaveowners. After all, the reason there was slavery in the South and not in the North is because the economy of the Southern states was dependent on plantation agriculture while the North was industrial. So there's no reason a Confederate victory would've prevented the spread of civil rights in what remained of the USA or the rest of the planet.
After all, the British abolished slavery before America did. It's a mistake to treat American history in isolation from the rest of the world. Even if the CSA had won the Civil War, it would've been under political and social pressure from other nations (including its neighbor the USA) that had abolished slavery. The global trend toward the eradication of legal slavery would've eventually encompassed the CSA along with everyone else, though it would've taken longer. Certainly there would've been no need to preserve the institution once there was electric or gasoline-powered farm machinery that could do the work faster and more economically than slaves could.
The primary ideology that led to the formation and secession of the CSA was nationalism, not racism. They saw themselves as a separate culture from the Northern states and didn't like the North telling them what to do, with slavery being one of the main issues of contention but, in their minds, just one facet of the larger problem. So there's no reason why an independent CSA would have to remain racist forever. After all, the civil rights movement had its roots in the South. In a post-slavery 20th-century CSA, there would certainly have been a homegrown civil rights movement. It might've taken longer, but there's no reason it wouldn't have kicked in by the 23rd century.
So the idea that Confederate victory in the Civil War = all-white spaceship crews in the future just doesn't follow.