This character would have been a contemporary of Alabama's Governor George Wallace who (in) famously said, "segregation today, segregation tomorrow, segregation forever." This meant, "Your social status of today is your status of tomorrow, and forever." A person who shared that mindset would care deeply about whether their family had or didn't have ancestors going back "forever," (e.g. to the Battle of Hastings).
The Battle of Hastings was arguably the most important event in (modern) English history. Most prominent English families had men engaged in it on one side or the other. Not being "represented" in what as arguably the "creation" of modern England was a source of concern to lineage conscious English families.
After he won, William the Conqueror compiled a list of major landholders in the so-called Domesday Book. Most of these landowners, by definition, had some family member fight at Hastings because of the feudal system. Not having anyone in the family represented there after so many generations of intermarriage signified a lack of feudal ancestors with status, at least to some.
A commenter reasonably observed that "not everyone considers 950 years ago to be 'modern.'" That's the way that most people on Stack Exchange would feel. But the question was why did the author depict a character from the middle of the 20th century Alabama as using the Battle of Hastings as a "frame of reference." Put another way, the Battle of Hastings (almost 1000 years in the past) had relevance to Alabamans in the 1960s because those Alabamans believed that it had such relevance.
Southerners were more likely to feel this way than Northerners because a larger percentage of (white) Southerners were of English descent. Also, a larger percentage of English Southerners were "gentlemen" settlers, as opposed to e.g. Puritans, and therefore even more class conscious than other Englishmen.