The 14th Amendment contains the caveat. We sign SOFAs (status of forces agreeements) with host countries to deal with precisely these sorts of issues.
I doubt that an SOFA overrides the Constitution.
And when you say "the caveat," I assume you mean the "subject to the jurisdiction" clause? If so, yes. The point is: where do you draw the line?
Though actually upon reading up, this was dealt with and cited in the Wong case...
Some say it was dealt with. The Supreme Court did discuss it.
The question is: is the discussion that exceeded the bounds of the case, binding law or dicta
. Dicta is basically "shit the court said that doesn't carry the force of law."
Wiki: A judicial statement can be ratio decidendi only if it refers to the crucial facts and law of the case. Statements that are not crucial, or which refer to hypothetical facts or to unrelated law issues, are obiter dicta. Obiter dicta (often simply dicta, or obiter) are remarks or observations made by a judge that, although included in the body of the court's opinion, do not form a necessary part of the court's decision. In a court opinion, obiter dicta include, but are not limited to, words "introduced by way of illustration, or analogy or argument". Unlike ratio decidendi, obiter dicta are not the subject of the judicial decision, even if they happen to be correct statements of law. The so-called Wambaugh's Inversion Test provides that to determine whether a judicial statement is ratio or obiter, you should invert the argument, that is to say, ask whether the decision would have been different, had the statement been omitted. If so, the statement is crucial and is ratio; whereas if it is not crucial, it is obiter.
IMO, if Trump does enact an executive order, it will be challenged, and find its way to the Supreme Court. I predict a divided Court.