The list in this answer is not comprehensive merely listing some of the leading SCOTUS cases on point, but the approach taken, analyzing the way the term is used in a context specific, case by case approach, rather than trying to ascribe a trans-substantive meaning to it in all contexts is, I believe the correct approach. There are 30 SCOTUS cases that use the term discussing perhaps 15-20 treaties and statutes, and hundreds of circuit court cases that do so.
It turns out that the term foreign national often paraphrases treaty language that is not identical when there is no ambiguity in the case before the court. The EO language is probably merely the product of sloppy drafting since the statute uses the word "alien".
In many contexts where "foreign national" is used to paraphrase the exact treaty language, the term is relevant because it is used in a context where a relationship between the person and their nation is at issue in some respect as in the VCCR below and the Public Vessels Act. So, in those contexts, the term would often exclude "stateless persons". But, there is really no way to know in the absence of context whether dual citizens are or are not intended to be included in the term.
The VCCR: Paraphrase
One place that the term "foreign national" is used is Article 36 of the Vienna Convention on Consular Relations (VCCR) which gives a foreign national a right to diplomatic assistance in the criminal justice process. See Sanchez-Llamas v. Oregon, 548 U.S. 331 (2006). The same section of the same treaty was at issue in Torres v. Mullin, 540 U.S. 1035 (2003).
In this context, a stateless person is clearly not a foreign national, as a stateless person has no ambassadors or counsels from the country in which the stateless person is a foreign national to seek recourse. The term "foreign national" also paraphrases the treaty language which actually uses the phrase "nationals of the sending state" (where the "sending state" is the country with diplomats in the territory of the "receiving state") which could in a plain reading include dual citizens. Specifically it says:
1.With a view to facilitating the exercise of consular functions relating to nationals of the sending State:
(a) consular officers shall be free to communicate with nationals of
the sending State and to have access to them. Nationals of the sending
State shall have the same freedom with respect to communication with
and access to consular officers of the sending State;
(b) if he so requests, the competent authorities of the receiving
State shall, without delay, inform the consular post of the sending
State if, within its consular district, a national of that State is
arrested or committed to prison or to custody pending trial or is
detained in any other manner. Any communication addressed to the
consular post by the person arrested, in prison, custody or detention
shall be forwarded by the said authorities without delay. The said
authorities shall inform the person concerned without delay of his
rights under this subparagraph;
(c) consular officers shall have the right to visit a national of the
sending State who is in prison, custody or detention, to converse and
correspond with him and to arrange for his legal representation. They
shall also have the right to visit any national of the sending State
who is in prison, custody or detention in their district in pursuance
of a judgement. Nevertheless, consular officers shall refrain from
taking action on behalf of a national who is in prison, custody or
detention if he expressly opposes such action.
The Trading With the Enemy Act: Paraphrase
The term was also used in a case discussing the Trading with the Enemy Act, as amended by the First War Powers Act of 1941, and Executive Order 9095, as amended, which allowed for an "Alien Property Custodian" to seize property of a foreign national (in this case a German) in connection with World War II. See Silesian-American Corp v. Clark, 332 U.S. 469 (1947). It isn't clear, however, that this case has continuing legal validity and relevance. And, the "foreign national" terminology in this case appears to be a paraphrase of the phrase "enemy or ally of enemy" that appears in the underlying text of the authorizing statute.
U.S-Mexico Extradition Act and 4th Amendment: Synonym For Alien
The term was used in the case of United States v. Verdugo-Urquidez, 494 U.S. 259 (1990), in which a Mexican citizen was kidnapped in Mexico by U.S. agents and tried in the U.S. on drug trafficking charges without regard to the U.S.-Mexico extradition treaty, which Judge Kennedy uses the term synonymously with "alien" in his concurring opinion in a case involving facts where any definition of the two terms applies.
Hague Service Convention: Concept Not Used In Treaty
The term was used in the case of Volkswagen Aktiengesellschaft v. Schlunk, 486 U.S. 694 (1988) interpreting the Convention on Service Abroad of Judicial and Extrajudicial Documents in Civil and Commercial Matters, Nov. 15, 1965 (Hague Service Convention),  20 U.S.T. 361, T.I.A.S. No. 6638 which governs service of process of foreign nationals to determine if it applies to service of process on foreign nationals within the United States. The treaty language of the Hague Service Convention itself, however, makes no reference at all to the nationality of the person served and depends entirely upon whether the person is served abroad in the territory of a signatory or not. Under the plain language of the treaty itself, a U.S. citizen in France has to be served pursuant to the Hague Service Convention, although a U.S. court would usually be in a better position to compel compliance from its own national than it would be from a foreign national.
In a nutshell the facts and circuit splitting legal issues presented by the service of process in a product liability case in a state court in the United States in Chicago were as follows:
VWAG, a corporation established under the laws of the Federal Republic
of Germany, has its place of business in that country. VWoA is a
wholly owned subsidiary of VWAG. Schlunk attempted to serve his
amended complaint on VWAG by serving VWoA as VWAG's agent.
VWAG filed a special and limited appearance for the purpose of
quashing service. VWAG asserted that it could be served only in
accordance with the Hague Service Convention, and that Schlunk had not
complied with the Convention's requirements.
FWIW, SCOTUS held that the treaty did not apply to service of process within the territory of the country whose courts issued the process.
The Public Vessels Act: Context Specific Usage
In United States v. United Continental Tuna Corp., 425 U.S. 164 (1976), per the official syllabus:
Prior to 1960, the Suits in Admiralty Act authorized suit against the
United States in cases involving vessels owned by, possessed by, or
operated by or for the United States, if such suit could have been
maintained had the vessel been a private one, and provided further
that such vessel was employed as a merchant vessel. In 1960, Congress
amended the Act by deleting.the latter proviso. The Public Vessels Act
authorizes suit against the United States in cases involving "a public
vessel of the United States," but bars such a suit by a foreign
national unless it appears that his government [96 S.Ct. 1321] allows
a United States national to sue in its courts under similar
circumstances. Respondent, a Philippine corporation, alleging
jurisdiction under both Acts, sued the United States to recover
damages resulting from the sinking of its fishing vessel after a
collision with a United States naval destroyer. The District Court
dismissed the complaint on the ground that, since the destroyer was a
"public vessel of the United States," the suit was governed by the
Public Vessels Act, that therefore respondent was subject to that
Act's reciprocity provision, and that, since there was no such
reciprocity, the suit was barred. The Court of Appeals reversed on the
ground that the suit, although involving a public vessel, was
maintainable under the Suits in Admiralty Act, as amended in 1960 to
delete the "employed as a merchant vessel" proviso, free from the
restrictions, including the reciprocity requirement, imposed by the
Public Vessels Act.
Held: Claims within the scope of the Public Vessels Act remain subject to
its terms after the 1960 amendment to the Suits in Admiralty Act, and,
since respondent's claim falls within the Public Vessels Act, the
Court of Appeals erred in concluding that that Act's reciprocity
provision did not apply.
Here the focus is on the relationship of the foreign national to a home state court which may or may not offer reciprocity. The statute is 46 U.S.C. § 781. The case at issue involved a Philippine corporation owned mostly by Americans, which was held to be Philippine nationality and the reciprocity of Philippine courts applied. Presumably in this case a dual national would count as an American and a stateless person would not.