I don't have any official knowledge, and there are probaly people who know mose specifics. But, looking back into old media guides and yearbooks, the Bradley athletic teams were referred to as "The Braves" after WW2, and their uniform warmups had a logo of a stately Indian chief profile that continued through the 50's. But none of the programs or media guides show any indication of a mascot until a child-like Indian cartoon creature named "Lil Brave" started to appear in the mid '50s and lasted into the early 70's. This Lil Brave was replaced by an indian character holding a tomahawk in the mid 70's until the early 80's, then by an Indian Chief's head profile again in the mid 80's.
There was also an Indian Chief character that appeared in the '60's and performed at halftimes of games into the late '70s when it became frowned upon. There was a very short-lived mascot, a student dressed up in a goofy Indian costume as Brad E. Lee in the mid '70s, but that only lasted 1-2 years, also giving way to pressure from native American groups. Then, of course, there was Tommy Hawk for a few years in the 80's and early 90's, I think.
Looking into prewar yearbooks, the teams were mainly referred to as simply the Bradley men or varsity team. Occasionally terms like B-men and Indians are used, but they do not seem to be a reference to an official nickname.