this is done by completing the first tournament in the Cabaret club minigame. In order to even battle the other champ you need to earn at lease 1mil yen after taxes. The other club will only raise a maximum of 1.5mil yen in battle so 1.3mil after taxes. As long as you raise more than 1.5mil yen during the battle you will beat them. Be sure to recruit as many hostesses as possible, raise your fan count, and level up your hosteses.
https://imgur.com/a/cfSHDc6 This is a map of all of the first leagues partner businesses.