If she’s serious, this would be an interesting case of whether or not the Nobility Clause applies.
No title of nobility shall be granted by the United States:
The title of Dutchess was not 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 foreign state.
The Dutchess is not a person holding any applicable office of profit or trust under the United States, and the Dutchess did not accept any such title or emolument while holding any office.
She would not appear legally barred, but she would probably be politically unacceptable. Her favorability might be difficult to detect.
There is no record of judicial precedent to review, but the information at the below link may be informative.
https://constitution.congress.gov/browse/essay/artI-S9-C8-3/ALDE_00013206/