While I think we could honestly debate the English language of that era the Framers refereed the “Law of Nations” in many of their writings. It seems clear that the reference in the Constitution is that of the one and only book of the Law of Nations.
The phrase itself was common before it was the title of Vattel’s treatise. It would be interesting to get a constitutional scholar’s take on it. I believe that there were conventions for writing titles of books, plays, poems, and so on, and I believe if they intended to reference a specific text, they would have made that very clear. They could have also used words.
I don’t believe the Constitution acknowledges any of the sources of its content, so I do not believe the Framers would have named a specific book about international law and not the many other prominent influences on their thinking. Why this one? I could probably find out if I spent the time on it, but it’s not on my “open question” list; in fact, I am 100% certain it’s not the book that is named but the concept of the law of nations, which was a term well known to them that represented a body of international law. I’d bet on it, actually, because I’m that sure.
I’m very short on time at present, so I must keep the reply equally brief. Suffice it to say, the title, Law of Nations, has been in use by numerous authors of legal treatises on the subject for many centuries. See for example any essays you may find regarding the Roman jus gentium, IIRC. Of particular note to the Founding Fathers are Vattel’s works and essays reviewing his works and those of Grotius. To make a long story short, Thomas Jefferson for one example alone owned copies of many of these important works dating back to the Greek republics; the Roman juris civiles; and latest French, Swiss, German, and English legal and ethical works as they authored the U.S. Constitution. They are catalogued in his library holdings and were acquired in part while he was Ambassador Pleni-potentiary in France to suplement the library being rebuilkt after the fire at Shadwell.
Vattel was relied upon extensively by Thomas Jefferson and the other Founding Fathers in part because it was among the most recent and comprehensive works which represented their concepts of natural law in government in opposition to what they viewed as unacceptable in contemporary British legal philosophy and practice.
The Law of Nations under various names in various languages has existed as a body of customs and bilateral and multilateral international relations throughout the thousands of years in which kingdoms and empires have existed. Written works regarding the Law of Nations have survived in whole or part from several millenia.
Thomas Jefferson and the other Founding Fathers utilized Vattel’s work and many of the other works in concert to assist the effort of authoring the Constitution.