I believe that the rebellion was crushed before they had any time to actually hold a real election. Form 1864-on there wasn't really much of a governemnt to speak of at all; everyone was just kind of looking out for their own.
The Confederacy in fact had regularly conducted elections throughout the Civil War. Davis was elected president in Feb. 1861 by the 1st Confederate Congress, and in Nov. 1861 he was elected by the voters, unopposed, in a Confederacy-wide election. Congressional elections were held in the various southern states in every year from 1861-65, and they were all as "democratic" as any election that had gone on before, and as "democratic" as those going on in the north at this time.
Bottom line--the Confederacy was a democracy. And the Civil War was a war between 2 democracies.