Well, yes, in a way - the Confederate States of America had a formal government located first in Montgomery, Alabama, and later in Richmond, Virginia. This government had a constitution, president, and elected representatives. There wasn't any need of a government in exile during the war, per se, since the CSA had a huge swath of territory south of the Mason Dixon line to operate (on both a military and civic level). There was no possibility of a CSA representative in the Senate or Congress, as any person in D.C. professing loyalty (and acting as such) to the Confederate government would have been arrested for treason. Lincoln's administration steadfastly refused to enter into any talks or negotiations with representatives from the Confederate government precisely because to do so would mean that he recognized them as representatives from a sovereign nation (which he did not). Therefore, official U.S. policy was to treat any civic representative from the CSA as an individual actively operating in a treasonous fashion against the United States, subjecting them to prison and trial before a military court.

Any position held by a CSA official would have yielded power only within the Confederacy itself, and not on any stage larger than that. The diplomats the Confederacy sent to France and England were largely ignored (and did nothing to bring foreign intervention into the mix), so to answer your question, no, these officials wielded no real power (outside of the Confederacy).

I hope this helps, and that I haven't misunderstood your question.

[Sources - Bruce Catton, 'Mr. Lincoln's Army,' James McPherson, 'Battle Cry of Freedom'] EDIT - CSA capitol was originally in Montgomery, not Mobile