There are others in the pre-swing era who could qualify as "producers" n the more modern sense of the term.
Joe Davis provided the entire package for record companies wanting to get in on blues recording.
He published the songs, hired the singers and accompanists, and even provided sound effects, if needed.
As a aide line, he sang heart-on-sleeve ballads for a couple of different labels in the late 1020's/
And his career continued well into the 1950's.
Mayo Williams out of Chicago went from his own Black Patti label (try and find 'em!) to Vocalion, and thence to Decca. Even as late as the late 1940's, he had his own Harlem label, ad was still producing blues and rhythm records.
Besides Ralph Peer, Victor had, at various times, Edward T. King, Nat Shilkret, Leonard Joy and Leroy Shield--all of whom served a function analogous to today's producer.
As far as engineers go, Victor had Raymond Sooy, who was the engineer for the first Victor sessions of the Original Dixieland Jass Band, Earl Fuller's Famous Jazz Band--and presumably others--out of New York. He was still with Victor as late as 1931, if not later.
And, at OKeh, Charles Hibbard is often credited with making OKeh among the best-sounding acoustical records of the time. He was with the firm at lest through the "Truetone" electrials. I don't know if he stuck with OKeh after they had been bought by Columbia and switched to the Western Electric system of recording.
And, let us not forget Orlando Marsh, whose experiments with elecrrical recording gve us sides by King Oliver, Jelly-Roll Morton and others.