The POD can be the south winning the civil war, it works better if it is late during the war, but matters little.
After winning the war the south begins its reconstruction. Due to the fact that most of the agricultural fields were destroyed during the war most of the reconstruction takes place within major cities, causing major migration of poor white farmers into the cities. This does not necessarily mean major industrial growth only urban growth. Within time no later than the 1890s the slaves are freed. Like in OTL some become sharecroppers and others move into the cities. Cities then develop similarly as in OTL where blacks move downtown and whites move towards the suburbs, but nevertheless there is some mixing going on. The importance of New Orleans the largest city in the CSA is important as unlike the rest it begins already as a center for liberalism.
The CSA snatching Cuba and Puerto Rico from Spain will also work on its path towards liberalism, and would give it more diversity later on. Like in OTL there could be a Puerto Rican diaspora were most live outside Puerto Rico in this case within the CSA.
Having a wide range of ethnicities in one country usually creates cultural hybridization, even if it is not necessarily racial. Even in OTL the most hybridized culture comes from the south. Ultimately hybridization create something spectacular around the early 1900s ... Jazz or a similar music genre injected with some Cuban rumbas or both as separate genres. The music will be enjoyed by everyone bringing aspects of black culture into white culture.
Follow this with movies the CSA develops its own movie industry, most likely based in New Orleans. Movies can have several effects but because we are going for the liberal end we can say the early Confederate Filmmakers are liberal artsy people who question some traditional values and bring it to the big screen. Think of early French and German cinema of OTL. European culture is also coming in, most importantly French modernism, and French fashion. Flappers (or similar fashion) in the 1920 followed by the beginnings of the sexual revolution. Because every action has an equal and opposite reaction, the CSA's conservatism creates an even stronger liberation movement. Women gain the right to vote in the late 20s followed by blacks in the 30s or 40s (although they will remain in "segregation")
With the discovery of major oil fields the CSA, primarily the west flourishes and grows economically at an unprecedented rate.
At this time there is also some migration from Mexico into the CSA primarily into Texas. There is also some European in the east migration though to a lesser extent.
I will assume that the CSA, like the USA of OTL, saw little fighting in WWI expect supply its allies and maybe and intervention at the end nothing like TL 191. However as the late 40s approach the CSA is entangled in a second major conflict. Causing a backlash in the liberal revolution in the 1950s nevertheless helped by further industrializing parts of the CSA. New Orleans, Richmond, Atlanta, Birmingham and Havana (largest city in the Caribbean) are by now major metropolis. Black populations have also flourished and benefit from the expanding sources of jobs. As more whites move into the cities blacks who have made a small fortunes can invest in the countryside. Creating a constant flow and movement of both whites and blacks into the cities and out of the cities. (If blacks live in the countryside and downtown the whites in the suburbs are in the middle of the whole).
In the 50s the CSA flourishes economically though it becomes increasingly more conservative. However by the late 50s with the invention of television the liberal media is able to access every home. At the same time a new radical genre of music arrives and we see the birth of rock and roll infused with some latin sounds from Cuba and Mexican immigrants further closing the cultural divide. Like Jazz CSA music is enjoyed world wide.
We also see the rise of the Civil Rights movement, a Martin Luther King equivalent will eventually happen. His assassination, if there is one, will become an outrage and the liberal media will make sure everyone is to blame. If there is no assassination he will continue to speak firing up the movement.
In the 60s we see something never seen before a European band brings European pop culture into the CSA. They are a sensation. Although it is clear they were influenced by Confederate music but have simply adapted it and made it their own, they speak to the Confederate youth, white black and hispanic alike, with an power never seen before and a message of peace and equality. The young generation tunes into their message and begins to experiment.
Throughout its history the use of drugs, primarily tobacco and caffeine, has been enjoyed through out the CSA. Some states had gone dry at the beginning of the century but it did not last long and only created a tradition of moonshine and smuggling, helping the liberal cause in the end. In the 60s and 70s we see a major increase in the use of marihuana. Hemp is now a major industry in the CSA, although its psychoactive use remains illegal, there has been large investment in its industrial growth to avoid importing lumber from the USA. Along with marihuana other drugs hallucinogens and cocaine begin to be used. Unfortunately this creates a major problem of drug trafficking, and the CSA is a major grower of most drugs due to its reliable weather. There is a major struggle to keep the drug use down but the youth seems not to care and rebels against it.
The CSAs national and historical identity appears to help very little. Historically, although conservative, they have seen themselves as rebels. They rebelled against England they rebelled against the USA, they helped Cuba rebel against Spain and thus allowed it to join. The new generation only seems themselves as the new necessary rebellion. A Liberal party grows in the CSA with the slogan "We Need a New Revolution!" this rings truth in the youth's minds. As part of the revolution the remaining segregation laws across the CSA are abolished, and the sexual revolution returns. In particular the efforts of a young and eccentric Texan help this effort when he creates a magazine that brings the pinup into the mainstream. The publication had existed for a decade now but after it is censored by the government the bold CEO sends his case to the supreme-court, arguing censorship limits the freedom of speech. The media turns towards his case and makes almost a big of a fuss about it. In 1968 the censorship laws across the CSA break apart. As part of this the production code on most of its cinema and television disappear and for the first time filmmakers tune their work for the excitement of the senses breaking most of the established taboos and shocking its audiences.
By the late 70s the storm has calmed down, an internal drug war sponsored by the conservative forces stopped most of the illegal trade within the CSA, and a rating system had been established as an alternative to censorship. However the damage was done and the CSA had become an increasingly liberal society. The youth that saw the 60s storms had children and the 80s ushered a new era of prosperity a boom. By now white was no longer the majority only the largest minority as more hispanic population immigrated northwards form South America into Cuba and Florida and from Cuba and Puerto Rico into the rest of the CSA. The boom of the 80s economy ended with the growing liberal party being elected into office with a majority on both houses for the first time.
Deciding not to ignore the truth, the psychoactive use of hemp is legalized sometime in the 90s. The CSA enjoys a bright and liberal future.
The End
For those wondering what happens with the USA. Well use Newton's law again, for every action there is an equal and opposite reaction. Unable to cope with CSA's success it becomes a military dictatorship suffers from coup after coup until it rebuilds itself in the 30s - 50s although broken and isolationist as it had always been before the CSA.
Although this is an exaggeration, my point is that it is very hard to predict the development of culture. Here in AH.com it seems we tend to focus on military and political developments but cultural ones are just as important they all affect each other. I think it is very plausible for the CSA to become liberal, it is just about how one generation reacts to the previous and having the music, and food (something i did not mention above) that the south has enjoyed through its history are definitely a push towards liberalism.