re: Can anyone Identify this military style LSU cap?

Could be. They did have drum majors after 1946 I believe is the date they broke away from ROTC but the uniforms remained military until 1958. They had drum majors previous but it was earlier before Huey P. Long was assassinated. After he was killed LSU went back to military and therefore had ROTC Band Officers.

This is an excerpt from the book Kingfish U:

In January 1931, LSU announced new band uniforms that maintained the unit's military bearing while introducing additional elements, including aiguillettes— a braided, gold cord resembling the French decoration for valor-which looped over the players' left shoulders. The military-style caps were now made of purple cloth and trimmed with gold braid. Around their coats, members wore a gold belt.*
110 | KINGFISH U

But this cap is later than 1930's.
See 1935 and 1945 (this one is close)But neither are it. I can't find any good pics from 1949-1958. [url=https://postimg.cc/sBRNYgBp] [/url] [url=https://postimg.cc/H82YghfN] [/url] [url=https://postimg.cc/H8HYFgP5] [/url]