Please can I try to help with a little more background information?
The British Royal family has encountered the problem of a Princess marrying a spouse without a title on seveal occasions: In 1973, when the Princess Royal was still known as Princess Anne, she married the then Lieutenant Mark Phillips (later Captain Mark Phillips]. After her marriage, the Princess was referred to as 'HRH Princess Anne, Mrs Mark Phillips' in the Court Circular etc
A decade or so earlier, Princess Alexandra had married the Hon Angus Ogilvy, who was the younger son of the 12th Lord Airlie. It was reported that Angus Ogilivy was offered a peerage but that he and Princess Alexandra had declined this. During the first part of her marriage, the Princess was known as 'HRH Princess Alexandra, the Hon Mrs Angus Ogilivy'. Her Husband was subsequently honoured by the Queen and made a Knight Commander of the Royal Vicotrian Order, ['Sir Angus Ogilvy] whereupon the princess was known as Princess Alexandra, the Hon. Lady Ogilvy.
Should Princess Beatrice marry her boyfriend, then, as HM Quee Catherine as said above, she will become HRH Princess Beatrice, Mrs David Clark. It is all part of the Uk law that a woman is entitled to the use of 'her husband's style and title', [whether her husband is a Prince or (merely) an Esquire]
Hope this helps,
Alex