Actually, the Confederates tried this. In many ways they tried to go back to the original vision.
I do see a way for it to work, but it probably requires an earlier POD, during the now mostly forgotten colonial period.
The framers of the 1787 constitution were not exactly trying to organized a democracy. Only the House of Representatives, among federal institutions, was directly elected. Not only were the Senators chosen by state legislatures, but the electors which elected the President were assumed to be as well. And there was no provision for one man one vote for the House elections, there was guidance on how House members were to be apportioned among the states, but nothing about how the states would organize the House elections. Districts of equal population size came about only in the 1960s, due to a Supreme Court ruling based on parts of the Constitution not instituted until after the Civil War, and federal legislation. However, it was provided that the franchise for the House elections would be based on the state franchises for the "most numerous branch" of the state legislature.
This was still considerably democratic by 18th century standards, but they this is by eighteenth century standards. However, many if not most state legislatures and colonial assemblies were at the time elected by a fairly wide franchise among non-slave adult men, though I think (I could be wrong) only Pennsylvania came close to universal adult male suffrage. So even these pretty restrictive conditions gave an opening for elections to be important. And once elections were important, you had parties to contest the elections, and an opening to widen public participation.
Note that Britain did not really have parties either, but they were starting to form to contest the few elections that were allowed, for the county MPs and for borough MPs where the boroughs allowed a fairly wide franchise. Of course the framers of the US constitution hated this. Except for maybe Franklin, who had lived in England, they don't seem to have made the connection that even allowing some elections, for some part of the government, would eventually lead for parties to form.
So what you do is to remove even the small element of democracy that existed in eighteenth century America. Either no colonial assemblies at all, though that will probably butterfly away independence in the first place, or all the colonial assemblies and state legislatures are chosen by a really restricted franchise, basically limited only wealthy property owners. There are elections, but only a handful of electors even in federal House districts. You don't actually have to change a word of the 1787 constitution to get this.
As a way of contrast, the colonial institutions of the Spanish and Portuguese colonies did not prove conducive for post-independence democracy, even in cases where the leaders of the new countries really did want something more liberal, and nineteenth century Latin American political party systems were rather sad.