Point is, the various sport associations of the empire were disunited from the get go because they were organised from below and the nationalist people never thought of creating imperial leagues and created in every case national leagues. Maybe if they were organised at least partly from above, like someone of the dynasty became an enthusiast seeing it as a chance to bring the empire together.
Someone who might be interested would be Franz-Ferdinand, he was also kinda federalist, so maybe promoting sport as a mean of union between the people would work. (Makes me think of Catherine de Médici by the way, if you can make to people dance together you can make them side together, peace by joy, politizing entertainments ...)
Probably only way is that Austria-Hungary somehow survives longer than in OTL (no matter how) and it federalise so that Hungary would be only just one state among Croatia, Bohemia, Galicia, etc.
No, this would on the contrary be opening Pandora's box. If Austria Hungary federalize AND every ethnicity that was under hungarian control remains under hungarian control, you'd have revolts everywhere, plus since Bohemia has it's own team, national team does not depend on the status of the nation, no matter if it's an actual state or something else. If you want to federalize, do it as United States of Greater Austria were planned, and then add in the contract "USGA is a country composed of many nations, who promise to work together and show a united face to the world." making diplomacy and external affairs national prerogative.