I think Calhoun could be a good choice. I don't think the Confederate founding fathers would name it after Washington (too confusing, what with that also being the name of the Union's capital.)
Maybe Calhoun or Leesburg or Leeville or something like that (Robert E. Lee would probably be more revered than Jefferson Davis. Or maybe Johnston after Albert Sidney Johnston, the great Confederate martyr.)
And if it were to become a federal district (which may bug some of the more anti-federalist Rebs) it would probably be D.D.: District of Dixie.