Allison --
I'm sure you're correct, and you're on the right track for researching this question.
The name of this camp not being familiar, I wondered if the name had been misinterpreted from handwriting on a military service record. After checking some other sources for regiments at Columbus MS in the summer of 1862, I find that it's correct, although spelling may vary. Unfortunately Camp Blewitt or Bluett isn't mentioned in the Official Records.
A map of wartime Columbus might help.
Best I can tell, the hospital/camp was about half a mile or less away from the depot in Columbus MS. For a couple of reasons I don't believe there were any buildings there. First, the name disappears after the summer of 1862. Second, soldiers refer to it interchangably as a camp and hospital. During the summer it would make some sense to quickly set up a tent hospital in an open area not far from the railway depot.
The location of Friendship Cemetery in Columbus MS may be a good candidate for the site of Camp Blewitt/Bluett --
http://www.scvcamp1220.org/friendship.htm
For what it's worth, here's an article from the New York Times about Columbus and its history.