Since the Titles of Nobility Amendment was never ratified and probably would not be revived even under these conditions, all the Constitution says is that the federal and state governments cannot grant titles of nobility and that employees and members of the federal government ("no Person holding any Office of Profit or Trust under [the United States]") cannot accept titles except with the consent of Congress. A private citizen not employed by the government has no such restrictions, and arguably neither would an employee of a state or local government. There also doesn't seem to be any requirement that people surrender any titles that they already had upon taking office.I think the Constitution prevents Americans from accepting foreign titles or from giving people fancy titles. It doesn't say anything about marrying anyone who ALREADY has those titles.
I think JFK Jr. can marry Princess Di, but he can't become, like, the Duke of London or a member of the Court of St. James.
Now, JFK Jr. probably wouldn't go fishing for any British titles since at the time of marriage he would obviously be running for President and might be soured on the idea by his wife, but he absolutely could do so if he wanted without running afoul of the Constitution.