By 1974, the Southern Strategy had been a mainstream political strategy of the Republican for arguably a decade, beginning with Goldwater's anti-Civil Rights Act campaign for President, which allowed him to win most of the South including Louisiana. Louisiana itself had not voted Democratic since 1964 at the time of the film, voting for Goldwater, then Segregationist Independent George Wallace, and finally Nixon in his 49-state landslide. However, despite the Southern Strategy being in full swing by 1974, with Nixon nominating "states-rights" supporting federal justices and even asking the Justice Dept to slow its implementation of desegregation policies, the Southern Strategy in the 1970's was not characterized by a mass switch in party affiliation or state politics in Louisiana. Democrats still controlled every statewide elected office and nearly every seat in the state legislature of Louisiana in 1974, and contemporary political scientists claimed that in 1974 the state was roughly 86% Democratic, and only 12% Republican in terms of voting behavior and party strength. Furthermore, the local bastions of the GOP in Louisiana were primarily in urban and suburban centers, not the rural parishes that are still willing to support local Democratic politicians even today. While White Southerners at this time were no longer averse to voting Republican as a result of the Southern Strategy, they still primarily conceived of themselves as proud Southern Democrats. Even as late as 2000 47% of Louisianans identified themselves as Democrats, in comparison to the 40% that identified as Republicans. While the Southern Strategy had successfully managed to allow for Republicans to win the Presidential votes of formerly solid Democratic Southern States and have limited breakthroughs in state politics, the process of changing how white southerners conceived of their political affiliations was a much more gradual process that had barely made an impact by 1974. Thus, the joke regarding JW Pepper's political affiliation can be seen as reenforcing his characterization as a stereotypical southerner. To politically aware audiences of 1974, Pepper being a Democrat would be expected, regardless of his racial attitudes.

Sources: Brown, Frank. “Nixon’s ‘Southern Strategy’ and Forces against Brown.” The Journal of Negro Education 73, no. 3 (2004): 191–208. https://doi.org/10.2307/4129605.

David, Paul T. “Party Strength in the United States: Changes in 1974.” The Journal of Politics 38, no. 2 (1976): 416–25. https://doi.org/10.2307/2129542.

Wright, Frederick D. “The Voting Rights Act and Louisiana: Twenty Years of Enforcement.” Publius 16, no. 4 (1986): 97–108. http://www.jstor.org/stable/3330161.

https://en.wikipedia.org/wiki/Political_party_strength_in_Louisiana [for information on the party composition of the state legislature in Louisiana over the years]