Why did the Cherokee Nation believe it could win its case in the United States Supreme Court

They wanted to exploit a loophole in american politics. Before them, the other Native people who ceded their lands had first been recognized as nations and then when the territory became their legally, they would give it away in deals with the American government. The Cherokee thus declared themselves to be a sovereign nation and not to give the land, which is why they took the case to the supreme court in order to be protected.