There're a fair number of interesting timelines which might be done involving a monarchical United States and just who the first of those monarchs might be. But that potential monarch needs to, in 1787, be able to credibly argue for their own faith and belief in the American experiment. (As, having just fought a revolution to rid themselves of one king who did not understand or appreciate their rights, the Powers That Be would not consider giving it to another unless they were utterly convinced that he was -- spiritually at the very least -- one of them.) Given that Louis XVI's children sons were not even school-aged yet when 1787 rolled around, I'd concur with Kirook at "absolutely not", at least in 1787.
And after 1787, you're not going to get rid of the Constitution's prohibitions on the creation of nobility and the preservation of a republican form of government without ASB-level ridiculousness, as it's not going to be gotten rid of except by force of arms. (As the most plausible candidates to so thoroughly defeat a young United States as to enable imposition of a whole new government would not be inclined to actually do so.)