The Stadium District is its own governmental body and is separate from the State of Wisconsin government. It is like, for example, the City of Brookfield. If the City of Brookfield defaults on a contract, the aggrieved party does not have a cause of action against the State of Wisconsin just because the City of Brookfield is in Wisconsin, or because the State of Wisconsin has given money to the City of Brookfield in the past.
If the State of Wisconsin is a contractual guarantor of the Stadium District I am not aware of it, would not assume they are, and if someone claims they are I'd like to see the evidence of it.