Relating these two stories to a shared depiction of (or commentary on) the South seems like a rather tenuous proposition. There are, however, some ways to draw a connection between the stories.
Violence is the most dominating theme that the two stories share. Though the violence in Ellison's story is blatant, it is driven by a deeper implied violence which animates the men who set up the fight. The willingness to exploit minority children for entertainment demonstrates a rather monstrous acceptance of or alignment with violence on the part of the men who set up the fight. There is a wickedness to the formula, inviting children to fight one another then compete for a scholarship with speeches.
The violence in Faulkner's story is similarly subtle and implicit but does not share the outward manifestation of violence with Ellison's story. Though a dead body is discovered in the end, the death is not depicted directly and murder remains only suggested.
Furthermore, Miss Emily's behavior demonstrates a willingness to exploit her standing in the community. She refuses to pay taxes and refuses to explain why she wants to buy the rat poison because she knows she can get away with it. Like the men who set up the fight in "Battle Royal", Emily is willing to use her privileged position for her own selfish gain.
Beyond violence and exploitation, there is also a connection to be drawn regarding the consolidation of power in the hands of small groups of individuals who yield that power in capacities that are public and private, official and unofficial.
Regarding the South in these stories, there is not much to say beyond the fact that each story is set there and explores certain cultural elements of the South. While "A Rose for Emily" can be seen as an expression of Southern mores and Southern identity, "Battle Royal" is not as expansive in its perspective, nor as representative in its efforts as it is focused on the experiences of one boy and narrated through him.