OOC: This is all extremely unlikely. The Union has had rough patches in its history--censorship is nothing new, and the Lincoln Election of 1864 had "supervised voting" in some regions. But it remained Democratic. I'll try to follow that line.
IC: The Federal System of American Democracy is stronger than you assume at first glance. To put it mildly, American Politics, even in the turbulent 1880s with a grossly enlarged army, still had democracy on many levels--city, state, and federal. Pope's "March on Washington" being called a coup attempt is probably inaccurate, although demanding the concessions that the Army gets veto powers over the Defense Secretary and control of its own defense spending could have weakened Democracy considerably. In the worst case, Pope weakens but doesn't break democracy.
TR is another odd case: Recall that the USA would become more and more tied to its business elite. A similar clique--a quasi-fundamentalist group called the Populists--attempted to create a state church and other things under William Jennings Bryan. TR, despite his rhetoric and his aggressive attitudes, probably would have kept democracy intact.
We need to remember that the USA very easily could have moved very left or very right in the 1890s and 1900s--but it probably would have remained a democracy. Someone needed to break Corporate Power, and La Follette was the man for the job, although privately I think TR probably would have been satisfactory. I would become concerned at the power of the Hanna political machine.
That's a particularly hard jab at someone attempting to play on the fame of TR! Franklin Roosevelt contracted Polio and essentially was too crippled to stand on two legs--nor did he ever become more than Mayor of New York City. That and the membership of the "Greater Destiny Movement" generally remained committed to democracy, although quasi-imperialist. It was an outgrowth of "Manifest Destiny" in the 1800s.
The US government was in no danger in most of these situations. The worst problem, actually, came later when Arnold Palmer came to lead the FBI and started cracking down on leftists who turned out to be innocent. The USA has had some rough times, but the fact that states MIGHT be able to secede and this had to be considered, as well as the fact that US politics has never been dominated by one particular group kept democracy working. In an odd way, the secession of the CSA had the effect of the federal government keeping a weaker role in the affairs of states--states might secede if they weren't happy with the outcome, and while secession didn't work out too well for the South, notice was duly served.
In this regard, a TR dictatorship would lead to the fragmentation of America. Or, much more likely, TR plays a much softer tune. This explains why the USA Federal Government continues operate quietly even to this day, and why we rely on State Guards and Localized forces. Its hard to imagine a Federalized Army, just like its hard to imagine a Federal Income Tax.