Such an interesting question! There is something else that no one has mentioned yet, ... the Southern setting also serves to emphasize the contrast between the Keller/Sullivan connection. Let's begin with simple location. We can begin by saying that Annie Sullivan comes from VERY far away, near Boston, Massachusetts, actually. You can't get more Northern than that!
In addition, I think Gibson was also willing to make a parallel in regards to stubbornness. The South was stubborn after the North won the war, ... even to the point of celebrating the death of Lincoln. The Captain was stubborn about the traditions of the family even when Annie insists otherwise (as evidenced by the time when Captain Keller desires for Helen to continue eating off of plates but Annie insists upon ceasing that detestable tradition).
It would be a reach, but one could also make the case that Gibson is contrasting hard-headed men and women. The Captain and Annie are actually so much alike in that regard that it is their similarity that causes them to butt heads most all of the time. In this way, one could make a case about the challenge of gender roles within the play.
The idea of "bumping heads"-- I was thinking along those same lines--that the Southern background of the Kellers helps emphasize conflicts in the play, between Captain Keller and Annie and also between Captain Keller and his son. And those conflicts contribute much to the characters' development. The fact that Annie had to fight Captain Keller (especially him) so fiercely to save Helen really shows the depth of her commitment and the steel in her nature and personality. And then there's also the contrast between the old days (and ways) and the new.
It may be, too, that in the South, there are strong family values and a definite pecking order. The father is the head of the family, the mother follows, and then children. Most of the time, parents make decisions together, but in times of desperation, what the father says, goes.
Annie, a brash, independent, educated woman from the North, bumps heads with the Kellers several times--especially Captain Keller, for control over Helen. Neither Helen's father nor mother want to give up control of Helen--they just want things to be different. It is a power struggle from the beginning, and it disrupts the lifestyle of the Kellers who are accustomed to Southern ways.
Usually with a Southern setting, there is great contrast with the slowness, the gentility, the civility, the indirect discourse, which may or may not be accurate. But is contrasts other ares of the country very well.