I am not aware of any precedents on point.
If I were a judge deciding the issue as one of first impression, I would hold that it is disqualifying. The U.S. Constitution, Article VI, Clause 3 states:
The Senators and Representatives before mentioned, and the Members of
the several State Legislatures, and all executive and judicial
Officers, both of the United States and of the several States, shall
be bound by Oath or Affirmation, to support this Constitution; but no
religious Test shall ever be required as a Qualification to any Office
or public Trust under the United States.
A title of nobility generally implies of duty of fealty to the higher aristocrat from whom the noble title is derived, which is generally publicly recognized by swearing an oath of fealty upon assuming a noble office (the ceremony in which someone is knighted by kneeling before a lord or king and tapped on the shoulders with a sword is the most familiar circumstance in which this oath is given).
The oath of fealty to one's lord in monarchies was the model for the U.S. Constitution's oath in Article VI, and the parallel but not precisely identical oath of the President of the United States, set forth in Article II, Section 1, Clause 8 of the U.S. Constitution which state:
Before he enter on the Execution of his Office, he shall take the
following Oath or Affirmation:-"I do solemnly swear (or affirm) that I
will faithfully execute the Office of President of the United States,
and will to the best of my Ability, preserve, protect and defend the
Constitution of the United States."
The oath of office required by Article VI is inconsistent with an oath of fealty that runs to a sovereign monarch (directly or indirectly) of another country than the United States. Historically, and in some countries now, even today, a noble title also constitutes an ex officio obligation to provide military service to one's liege upon request.
This analysis is supported by the law of relinquishment of citizenship and naturalization under U.S. law:
8 U.S.C. § 1481(a) explicitly lists all seven potentially expatriating
acts by which a U.S. citizen can relinquish that citizenship.
Renunciation of United States citizenship is a legal term encompassing
two of those acts: swearing an oath of renunciation at a U.S. embassy
or consulate in foreign territory or, during a state of war, at a U.S.
Citizenship and Immigration Services office in U.S. territory.
The other five acts are: naturalization in a foreign country; taking
an oath of allegiance to a foreign country; serving in a foreign
military; serving in a foreign government; and committing treason,
rebellion, or similar crimes.
Beginning with a 1907 law, Congress had intended that mere voluntary
performance of potentially expatriating acts would automatically
terminate citizenship. However, a line of Supreme Court cases
beginning in the 1960s, most notably Afroyim v. Rusk (1967) and
Vance v. Terrazas (1980), held this to be unconstitutional and instead required that specific intent to relinquish citizenship be
proven by the totality of the individual's actions and words. Since a
1990 policy change, the State Department no longer proactively
attempts to prove such intent, and only issues a Certificate of Loss
of Nationality (CLN) when an individual "affirmatively asserts" their
relinquishment of citizenship.
So, while I am not aware of anyone who retained a noble title holding U.S. public office that requires that an oath of office be sworn, either under a statute, or under Article's II or VI of the U.S. Constitution, the better interpretation is that such a person is disqualified from holding such an office until the noble title is renounced.
The logic in the singular case of a sitting monarch taking an appointment isn't quite the same, because a monarch doesn't necessarily take an oath of fealty to anyone (although sometimes a coronation ceremony requires a monarch to take an oath of allegiance to the country he or she is about to rule). But there is a strong implication that someone who is the personal embodiment of a foreign state as its monarch cannot serve in a public office in the United States, with U.S. Constitution, Article I, Section 9, Clause 8 relating to Titles of Nobility and Emoluments cited in the OP certainly informing that reading even if it doesn't specifically say so in exactly those words. It states:
No Title of Nobility shall be granted by the United States: And no
Person holding any Office of Profit or Trust under them, shall,
without the Consent of the Congress, accept of any present, Emolument,
Office, or Title, of any kind whatever, from any King, Prince, or