Roger Kimball's crimes became public knowledge right at the end of TR's second term, so here TR will actually have time to do something about it. TR will certainly demand Kimball's extradition to the United States on threat of war if the Confederates refuse, so I imagine President Semmes will have no choice but to comply with TR's demands. Confederate voters will undoubtedly be upset about seeing their government cave to the US once again, so that might lead to an increase in support for the Freedom Party.
If that increase in support is enough for Featherston to actually win the 1921 Confederate presidential election, then the US and CS probably come to blows pretty quickly. In the original story, Featherston campaigned on unilaterally ending CS reparations. He'll probably have to do the same here. (The Freedom Party doesn't really have any other plan for dealing with Confederate hyper-inflation other than unilaterally ending reparation payments, so Featherston doesn't have much to offer the voters if he doesn't promise to end the reparations.) If President Featherston actually follows through on cancelling reparations, TR will presumably order an attack on the Confederates. That likely ends with Featherston getting overthrown by a military coup, and the US probably taking another chunk of Confederate territory. (Most likely Tennessee.) That will leave the military junta running the CSA pretty unpopular (they overthrew the democratically elected president and then surrendered territory to the US), so the CSA will probably be rather unstable thereafter.
Domestically I imagine TR will probably be able to come to some sort of accord with the Socialists in Congress once they realize they are stuck with him for the next four years. Probably something along the lines of he'll push for some of their domestic policy priorities (most likely old age pensions, unemployment insurance, and greater business regulations) in exchange for them going along with his foreign and defense policies.
What would TR's foreign policy goals be for a third term? Well aside from taking a hard line on the CSA, I agree with Lavanya Six that he'll probably get enough support to the rebels to help them win the Mexican Civil War. He's also likely to put the squeeze on Colombia or Nicaragua to get a US controlled Central American canal built. (Like Nivek said, TR will be leery of the Japanese, so he will want a US controlled Central American canal to enable US naval assets to quickly move from the Atlantic to the Pacific.) TR may also try to persuade the Danes to sell the Virgin Islands and Greenland to the US, though I'm not sure he'll have as much success there. (Germany may not react well to the US strong-arming Denmark.)
