Would sports (both in the sense of popular and lucrative) come to mean team games as exclusively as it did IOTL? How far along were the elites in buying into the 'playing fields of Eton' ethos?
The rural CSA would have some traditions of its own that could be used in building identity. IIRC horse racing and steeplechasing were big among the elites, while riflery contest, wrestling, boxing and cockfighting were more broadly popular diversions. A lot of the CSA is good horse country, where raising racehorses would carry far lower opportunity costs for the landowning gentry than in England. Equestrian sports could also become a popular cultural link to the western parts of the country. Cockfighting might even be embraced as something uniquely Southron and attain a controversial status similar to bullfighting in OTL Spain.
The CSA would also, of necessity, be a highly militarised society. They would need the firepower to defend themselves against a vengeful north, the strategic footprint to keep unwilling fringe territories under the thumb, and a militia force to quickly and decisively deal with slave insurrections. That makes it likely that every white male of military age would not only get some training - and more the more complex and physically demanding the modern battlefield gets - but would eventually be required to participate in reguilar exercises and/or musters. I envision something of a bastaredised Swiss model, where the federal government maintains fortress artillery and a navy of sorts, the states have standing forces ("permanent militias" or "state guards" or whathaveyou) and all citizens are liable for duty at need. That would certainly encourage competitive behaviour in related activities: track and field, martial arts, riflery &c. It would be in the interest of the states to organise and popularise such competitions.