As the Senior 1st. LT. Dr. Greenville Cook did command the company in the absense of Captain John L. Bradford. At one point Captain Bradford was absent from the company on detached duty and later absent recovering from wounds received during the siege at Port Hudson, at the war's end Capt. Bradford was a POW and did survive the war.
There are over 100 cards in Greenville Cook's military service records including copies of his hand written letters, copies of these records should be an added supplement to the family story.
Copies may be procured here -
Hope this helps,