Isn’t President Trump and other donors paying for the ballroom and the Country is not?

Is there a statute that grants the President the authority to accept outside donations and use the proceeds to construct facilities on federal property?

Under our system of government, the President's powers are limited to: i) executing the laws passed by Congress; and ii) the other powers specifically granted by the Constitution (to appoint officers of the U.S., to make treaties, etc.). If there isn't a statute that says the President has the power to do something, he generally doesn't have the power to do it (again, absent the Constitutional authorities).

The judge concluded that Congress has not given the President the authority to do this. They obviously haven't appropriated funds to do it, and they haven't given him the authority to accept outside funds in lieu of an appropriation to do it. Since the judge has determined that the President isn't executing an existing law in doing this, and since there's no independent Constitutional grant of power to do this, then it's outside the scope of his authority.