The National Archives service files nearly always include organizational records for each command. The first card usually lists field and staff officers identified as serving with the regiment or battalion. Many years ago I transcribed all those pertaining to Alabama, reviewed service files for each officer listed and made notes. If you're interested I can post assignments for the 38 Alabama medical officers serving with Archer's, O'Neal's, Law's and Wilcox's Brigades during the June-July 1863 time frame.

I had missed Assistant Surgeon William E Pearson, assigned to the 5th Alabama Battalion, and will add him to my notes. [Thank you!] Battalion records show him present for duty, July 1-3, 1863, so that makes the total 39.

Your list includes these officers:

9th Regiment: Surgeon Dr. W. F. Richardson. Richardson also worked at Camp Letterman.
4th Regiment: Assistant Surgeon John B. Wortham-present?
15th Regiment: Dr. Reed
47th Regiment: Surgeon John Robard Dutton (?) present

Asst Surgeon William F Richardson, appointed Nov 18, 1862, to rank from Sept 13, 1862. Richardson was captured at Gettysburg but reported his assignment as 2nd Florida Regiment. His prisoner of war records are filed with that command.

Asst Surgeon John B Wortham originally belonged to Co F, 4th Alabama Regiment, but never served as a medical officer with that command. He signed a receipt at Winchester VA, July 14, 1863, for a "medical train" including four horses. Wortham volunteered in 1861 as a resident of Huntsville AL.

Cannot find documentation for a Dr Reed serving with the 15th Alabama Regiment.

Cannot identify John R Dutton as a Confederate soldier or officer. You probably meant John R Burton, Surgeon of the 47th Alabama Regiment. He's one of the 38 medical officers mentioned earlier.