Is it possible that once Charles becomes King, and writes new Letters Patent, he could strip Beatrice and Eugenie of their Princess style, and have them styled as Lady?
Yes, it is possible.
The title of a British Prince or Princess is entirely at the will of the Sovereign and can be granted or revoked at any time. If Charles decides to downsize the Royal Family, he can issue new Letters Patent whereby Beatrice and Eugenie will be styled and titled as children of a Duke - that is to say, Lady Beatrice and Lady Eugenie.
In all probability, the changes would not be directed against the York Princesses personally but will concern the entire Royal Family. For instance, Charles could issue Letters Patent limiting the style of Royal Highness and title of a Prince/Princess to the children of the Monarch and children of the Heir Apparent to the Throne (in his own reign, that would mean William, Harry, and William's - but not Harry's - children).
That has happened before: when George V issued his Letters Patent of 1917 specifying who was entitled to the title of British Princes and Princesses (children, male-line grandchildren of the Monarch, and the eldest son of the eldest son of the Prince of Wales), several royals who had been British Princes and Princesses were ones no more.
Those royals were:
- John Leopold, Hereditary Prince of Saxe-Coburg and Gotha (great-grandson of Queen Victoria)
- Prince Hubertus of Saxe-Coburg and Gotha (great-grandson of Queen Victoria)
- Prince Alastair, 2nd Duke of Connaught and Strathearn (great-grandson of Queen Victoria)
- Prince Ernest Augustus IV, Prince of Hanover (great-great-great-grandson of George III)
- Prince George William of Hanover (great-great-great-grandson of George III)
- Princess Sibylla of Saxe-Coburg und Gotha (great-granddaughter of Queen Victoria)
- Princess Caroline Mathilde of Saxe-Coburg und Gotha (great-granddaughter of Queen Victoria)
- Princess Frederica of Hanover (great-granddaughter of George III)
- Princess Marie Louise of Hanover and Cumberland (great-great-granddaughter of George III)
- Princess Alexandra of Hanover and Cumberland (great-great-granddaughter of George III)
- Princess Olga of Hanover and Cumberland (great-great-granddaughter of George III)
- Princess Fredrica of Hanover and Brunswick-Luneburg (great-great-great-granddaughter of George III)