I think there's a key point that's been implied but not stated: members of the peerage use their highest title, and a Royal Dukedom title outranks a non-peerage Prince title. The one exception is territorial -- when HRH The Prince of Wales is referred to as The Duke of Rothsay when in Scotland, for example.
This underlyes what Curryong was saying when she explained that it would be rude to go by Prince Harry if the monarch bestows a Royal Dukedom on him at his marriage.
If you look at official royal communications, titles like Prince Charles, Prince William, etc are never used. The KP twitter will refer to William as, first, The Duke of Cambridge and then, in subsequent tweets about the same event, either HRH or The Duke. If you look at this press release about Anne (
https://www.royal.uk/princess-royal-visits-peru) she is referred to as The Princess Royal, The Princess, and HRH. Never Princess Anne.
"Prince/Princess Name" is used by newspapers and people because it's familiar, but it's not their appropriate title anymore, once they're granted a higher one.
I'd actually argue that for William, for example, it's not correct to say that it's "Prince William, Duke of Cambridge". He's The Duke of Cambridge, whom people happen to call Prince William. (This standard colloquial use is one of the reasons there's so much awkwardness around Catherine's naming, because The Duchess of Cambridge is a mouthful, "Princess Catherine" is just wrong, and "Princess William" is just awful.)
(Also, it's why it was rude of the Daily Mail to refer to Joe Biden's wife as "Mrs. Biden" in the article about tonight's basketball, since Dr. Biden holds a doctoral degree -- it's her highest title, not Mrs.)