In this I agree with you. The reference in the US Constitution to the “Law of Nations” is generic, i.e. non specific to Vattel. There were several other writers on the Law of Nations, among them Grotius,Puffendorf and Locke.
I agree that the Constitution citing ‘law of nations’ in lower case (my copy) would substantially indicate the Founders, who were scholars, had in mind a general application for that Sec. 8 of the Constitution. However. it is a recorded certainty that Franklin had copies of Vattel’s works which he shared with others including members of the Congress. Taking all recorded events together with Jay’s suggestion to Washington it is to my thinking that Vattels words were the/a foundation for the word ‘natural’ put into the Constitution. One thing is important and that is that by Vattel or any other writings ‘natural born’ is a unique requirement for a document such as our Constitution.