I’m no lawyer, but, I think that Article VI is clearly stating a hierarchy of laws and authority. The Constitution trumps Federal laws, which trump treaties, which trump State constitutions and laws. I think that proof of treaties’ inferiority to Federal law can be shown by the fact that no Indian tribe, AFAIK, has ever been able to recover damages when Congress abrogates a treaty. (They’ve had to make any recoveries the good, ol’ fashioned way: By bribing Congress!) If your theory were true, the Senate and the President could conspire to amend the Constitution by entering into convenient treaties with any banana republic wanting a little cash.
No treaty has ever been trumped by the Constitution in the Supreme Court. If the example provided in the Convention on Nature Protection isn't enough, I don't know what is.