Since Prince Harry, Duke of Sussex, is my 8th blood cousin 4 times removed on my father's, Louis McClanahan, family side, I would have no objection.
Currently when I am discussing Prince Harry, I will address his name as Prince Harry, Prince Harry, Duke of Sussex, and/or HRH Prince Henry Charles Albert David of Wales, Duke of Sussex..
no. direct heirs to the throne do not have a surname
Why Queen Elizabeth and the Royal Family Don't Have Last Names
http://time.com/5219397/royal-family-queen-elizabeth-last-name/