David --
You wrote, "I believe King enlisted as a 3rd Lt. in the 62nd, which took senior enlisted men and promoted them to officer of the cadets." As mentioned in another post, King was never an enlisted man after his appointment as 2nd Lieutenant, PACS. I don't have a record of his appointment, but it would have been in effect as a "permanent" rank with the War Department. Most volunteer officers did not have a commission in the PACS.
Also, the 62nd Alabama did not exist until a short time before the war ended. Since King received pay as Sergeant until mid-September 1862, we have to assume his commission took effect at that time.