In short, you're not going to accomplish it without a pre-1900 POD.
The Republican Party dominated national political power from 1860 until 1932. While Democrats occaisionally took control of Congress or the Presidency, they generally did so only by exploiting weakness or internal division within the ruling Republican coalition. Under these conditions, the Republican Party became the natural party of government, closely tied with business and economic interests.
Additionally, the continuing legacy of economic and cultural sectionalism effectively made the Democratic Party the regional party of the South. From 1876 until 1932, we really shouldn't even consider the Democratic Party a single party, but rather two separate wings sharing the name in a cynical alliance. As the Republican Party became the party of the established economic interests, the Northern wings of the Democratic Party became the primary venue for dissident liberals and economic populists.
This arrangement was set in stone by 1900. For the Bourbon Southern Democrats to break it would be to surrender any hope for national political relevancy. They were wedded to the arrangement even as the Northern wing began to overpower them (from 1912 to 1918 in particular). FDR and the New Deal were simply the culmination of a decades long political trend.
The Civil War made the Republicans the natural party of governance. To compete, the Democrats had to become the party of populist and liberal interests in the North. You can stop this trend, but it won't entail the Republicans becoming a mirror image of what they are IOTL. Northern workers and liberals occaisonally flirted with breaking their electoral alliance, such as with the Populists, the Socialist Party of America, and later the abortive attempts of establishing Progressive or Farmer-Labor parties from the teens into the 20s, but those efforts failed.