The Guide to the Papers of Charles Culmer, 1941 - 1944 does offer some clues if you read it carefully.
1. In the Introduction to the Collection:
"Of note is a history of the base begun by Culmer and carried on by subsequent officers.", which suggests he left the base at some point.
2. In the list of reports & manuscripts all have an end date of 1943.
3. The only thing with a date of 1944 is listed in:
Series 3, Personal Correspondence, 1944
"Personal Correspondence includes a Christmas card sent to Maj. Culmer in Minnesota from Maj. Hassell in Presque Island (mentioned in the Post-diary Report as the pilot who brought supplies and mail to the base)"
The christmas card was sent to his home in Minnestota so he must have left the base before December 1944.
The 15th Evac Hospital wasnt inactivated in Italy until 8 Sept 1945, plenty of time for him to be sent to Italy.
Found a book on Amazon.com, might have some info of use.
A surgeon's diary: 15th Evacuation Hospital : experiences in World War II, North Africa -- Sicily -- Italy by Frank W Peyton (1987