The enlistment date and place cited (July 4, 1861, San Antonio TX) match the roll for E. Burke, Co. "D", 2nd Alabama Cavalry. Burke served for over a year as a member of Mechling's Texas Battery, but you have already seen those records. Being from Alabama, it makes sense for him to have requested and received a transfer to the Tuscaloosa cavalry company.
Burke could have done additional work as a signal officer with the 2nd Alabama Cavalry, but there are no extra pay records to show it
Service files indicate that Burke remained with the 2nd Alabama Cavalry on the Florida border until May 24, 1863, the date he accepted his commission. By the time the War Department acknowledged his acceptance, A. P. Hill's Corps would have been marching west and north from Fredericksburg VA. Burke would not have arrived at Heth's or Hill's HQ until orders arrived directing him to do so, and the Army of Northern Virginia was hundreds of miles away. I see no reason to believe that Burke served north of the Potomac River.
A signal officer had important responsibilities. A. P. Hill's youthful record aside, if personal problems make it difficult for this officer to do his job, Heth might have simply lost patience with Burke.