I'm afraid I have nothing to offer you.
Its my understanding that the Nashville National Cemetery is for Union dead. Having said that, the Gettysburg National Cemetery was set aside for Union dead as well, but recent research has uncovered a small number Confederates buried in the national cemetery there that were misidentified as Union when buried.
The Marietta Confederate Cemetery would be the probable burial place for Richard B. Pittman, though its certainly possible not all the Confederate dead were disinterred @ Chickamauga, and could very well be there today.
Have you contacted the park service historian @ Chickamauga to see if he or she has a file indicating where the dead of Patton Anderson's brigade were buried on the battlefield?