The Chicago Blackhawks can restore the space between "Black" and "Hawks" and change their logo to a literal black hawk (as in, the bird).
Meanwhile, the Atlanta Braves of MLB can acquire the rights to the identity of the former NHL team Atlanta Thrashers and use it verbatim: name, colors (with Thrasher Ice Blue now being called Thrasher Sky Blue), and logos (with the hockey stick of the original main logo being changed to a baseball bat).
Atlanta Thrashers - The (unofficial) NHL Uniform Database
The four uniform designs used by the NHL team over its short existence would be the basis for alternate uniforms for the baseball team. There would also be Capital Copper and Peachtree Gold alternates.
Plus with my proposal to move the Atlanta MLB franchise into the Continental League as part of a
history-based realignment of MLB, it would have the franchise go further in cutting ties with its past as the Braves name would solely be an element of National League history. It would also mean that all of Atlanta's big 4 teams are named after (and use logos based on) birds (Falcons, Hawks).