In one photo of the Lake Forest Orchestra, 1922, Bix holds his cornet in his left hand. There seems to be no choreography in this photo regarding positions of the instruments. (Evans and Evans, p. 101.)
On the next page there is a photograph of a larger orchestra group in which Bix is holding his horn in his right hand. Again, there seems to be no pattern as to how instruments were held; some people have instruments evident and others do not.
In the 1924 shirtsleeves and vest photo of the Wolverines at Gennett (Evans and Evans, p.124) Bix splits the difference, holding his cornet across his lap with both hands.
In the Rhythm Jugglers photo of 1925 at Gennett, (Evans and Evans, p. 180), the cornet dangles from the right hand of the arm thrown over Don Murray's shoulders.
And the famous later professional photo of Bix with his cornet held high near his face shows it in his right hand.
I think we've pretty well established that Bix was ambidextrous when it came to cornet-holding.