Dr. William L. Lipscomb does not mention a Camp Bluitt/Blewitt in his book, A History of Columbus, Mississippi during the Nineteenth Century, but he does give several pages credit to a generous benefactor of Columbus, Thomas G. Blewett, Sr.

Although Dr. Lipscomb gives the year as 1862, Thomas G. Blewett's youngest son, Capt. Randle Blewett, having raised a company to take part in the great Confederate war, his noble and patriotic father generously armed, uniformed and equipped the whole company at his own expense only to see his gallant boy fill a soldier's grave in one of the great battles of Virginia.
(Apparently this is a memorial headstone in the Friendship Cemetery, Columbus, MS)
http://www.findagrave.com/cgi-bin/fg.cgi?page=pis&GSln=blewett&GSfn=randle&GSbyrel=all&GSdyrel=all&GSst=27&GSob=n&GRid=12915293&PIgrid=12915293&PIcrid=187885&PIpi=3282499&

From the above, I'm inclined to believe Camp Blewett was the correct name.

See this link for a list of hospitals in Columbus following the Battle of Shiloh:
http://www.rootsweb.com/~mslownde/Possumtown/Column_2.htm