Earlier, I had what seemed like a great idea:
Personally, I would be prepared to read the words “the United States shall exercise complete jurisdiction and control” language of the treaty as invoking the powers of all three branches of government, not just the executive. In this view, under Art. VI of the Constitution (“This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any state to the Contrary notwithstanding.”), the treaty would supply the jurisdiction for the federal courts that they seem to believe they lack under Article III.
On reflection, this isn't quite as great an idea as it seemed. Or, maybe it is, but it isn't as simple to get there as I would like.