“Since the US Constitution refers to the ‘Law of Nations’”
No, it does not. It refers to “the law of nations,” an ancient concept which is in no way limited by Vattel’s book, yes. But that phrase meant no more than “international law” means to us. If someone had years ago written a book called “International Law,” you certainly wouldn’t be under any compulsion to assume that’s what was being cited whenever the phrase popped up in current law. Which undoubtedly it often does. Because, like “the law of nations,” it’s a catch-phrase.
Besides, if one refers to "International Law", I'm sure that most of it is codified by multiple treaties, i.e. Geneva Convention, etc. So, while it is a "catch-phrase", the founders has some definite ideas what it meant.