Domestically, I think Nivek is correct. Roosevelt will probably reach some sort of understanding with the Socialists where he helps enact some of their domestic priorities in exchange for greater defense spending and a free hand on foreign policy. Thus the old age pensions and unemployment insurance bills that were successfully fillibustered by the Democrats in the mainline TL-191 verse, likely get passed ITTL (which should soften any subsequent economic downturn considerably.)
On foreign policy, I would expect Roosevelt to have the U.S. help the Republican faction win the Mexican Civil War. He might also push for a U.S. controlled trans-oceanic canal in Central America.
As for U.S.-C.S. relations, a reelected Roosevelt will certainly make the Confederate government extradite Roger Kimball to the United States. Turning over a highly decorated "war hero" like Kimball will make the Whigs look really weak going into the 1921 election, so that might be just enough to get Featherston elected that year. If that happens the U.S. and C.S. are certain to quickly come to blows. Featherston campaigned in 1921 on a platform of unilaterally ending reparation payments to the United States, so as President he pretty much has to follow through on that or immediately become a lame duck, and TR isn't going to stand for the Confederates canceling their reparation payments. I would expect the U.S. to respond to Featherston cancelling reparations by occupying Tennessee and/or Cuba which will probably lead to the Confederate Army overthrowing Featherston and establishing a military government that will then come to terms with the United States. (General Stuart already despises Featherston, so if he thinks Featherston is leading the CSA into a hopeless war with the United States, he will probably take action.) If there is such a coup then the Confederate States will be extremely unstable in the aftermath, with civil war or one or more of the states attempting to secede being real possibilities.
The novels had Roosevelt die of natural causes in 1924, so it is unlikely Roosevelt lives through his third term. The sympathy factor might be enough to have Roosevelt's veep win reelection, but I think it is more likely the Socialist candidate (likely Sinclair again) wins in 1924.