I am not certain if they were competitors, but here are some bands.
In his magnificent book, The Jazz State of Indiana, Duncan Schiedt gives accounts about several bands in Indiana.
The Crimson Sernaders and Hoagy's band played at Indiana University in the early 1920s. Charlie Davis had a band at Notre Dame. I believe most schools had a band, mostly dance bands rather than jazz bands, but they could do a bit of hot playing.
Duncan mentions the dance hall "The Green Lantern" and several bands played there, Ross Franklin, Ray Stillson, the Coon Sanders Nighhawks, Arnold Johnson, the Royal Peacocks, Coonie's Conrad Collegians. At the Casino Gardens in Indianapolis, they had the Miami Lucky Seven and Hitch's Happy Harmonists.
Other bands were The Riley orchestra of Muncie, Connie's Riverside Orhestra, Emil Seidel, Hal Denman and the Carolina Cotton Pickers,
Chicago bands mentioned in Timeless Early Chicago Jazz are Albert Short, Paul Biese, Joie Lichter, Joseph Gish, Art Kahn.
Maybe others can come up with additional groups who represent better comparisons with the Wolverines.