"At any rate, the Democrats' two bases -- Southerners and union "hard-hats" -- have always been more socially conservative than the cosmopolitan suburbanites in the GOP"
Southerners yes, but hard-hats? I'd say it wasn't true until roughly the 1960s. Before then, everyone was socially conservative, and the hard-hats if anything were more likely to thumb their nose at the church or "middle class morality," ect. It was only in the 1960s when the upper classes began to embrace feminism, homosexuality, ect, that the hard-hat reputation for social conservativism began. I would say a good POD would be in the early 1900s if you assume that the Catholic-dominated northern branch of the Democratic party began to take more control of the party from its Southern Protestant counterpart. By the time the Democrats nominated their first Catholic presidential candidate sectarian tensions had died down, but imagine an alternative time line when a Catholic Democrat is nominated in the early 1900s, how many votes would he get in the South? What if every nominating convention became a battle between Catholic and Protestant? You could then imagine Northern and Southern protestants uniting in reaction, under the Republican banner.
Another potential POD concerns the Civil Rights Issue. This was an issue that solidified support for the Democrats in the South, but what if it was the Republicans which were the less pro-civil rights party? You don't have to have them opposing it, the northern branches of both parties wanted it, the Democrats simply had less enthusiasm. Maybe, as part of the general Protestant-Catholic split, Northern Protestant Republicans would start to see their fellow Protestants in the South as valuable allies, with a similar problem of an alien population.
A complicating factor would be the Blacks in the South, whose historical attachment to the Republican party, and revulsion for the Democrat, has long been noted as a force resistant to change. However, I could see them potentially moving to a third party if the Republicans were more lackluster about appealing to them. It had been, from around 1880 to 1950, a somewhat illogical alliance, the richer, more protestant northern whites aligned with the poorest community in America. Perhaps they found neither party welcoming to them, so instead create a third party.