Interestingly enough, just as we were talking on the other thread, someone on this thread made reference to a ruling that I hadn't heard of: Reid vs Covert. It doesn't quite rule a treaty unconstitutional, in terms of the obligations it imposes on us, but it does say that a treaty can't authorize the U.S. government to do something that the Constitution itself doesn't allow. The subtle difference here is that the provision of the "treaty" (actually it was an executive agreement) in question only conferred a right upon the U.S. side, not an obligation to the other signatory (G.B.). IOW, it didn't matter to G.B. whether we exercised that power or not, because it was a concession that they made to us, not the other way around. Still, the language of the ruling seems to show some promise with regard to situations where a treaty (such as the migratory bird treaty) purports to impose an obligation upon us.
Just to follow up on the point I made earlier about the two-thirds requirement, I do know that we had a very harrowing experience prior to the adoption of the Constitution when it came ratifying treaties. Under the Articles of Confederation, a treaty had to be ratified by no less than nine states, period. But for whatever reason, only seven states were maintaining delegations in Congress when the peace treaty with G.B. was being brought up for consideration, and there was very serious concern that if it was delayed too much longer, the war might resume, and that could likely have been the end of everything for us. Eventually, of course, we were able to get two other states to send delegates so it could be ratified, but it was hairy for awhile there. So it does appear that I was wrong earlier when I speculated that the convention had assumed that most senators would be present at the seat of government at the time a treaty was being considered.
BTW, if I was a betting man, I'd bet money that Padilla will get his day in court - a civilian court.