From BBC News archives:
On 29 August 1950, the name of the new princess was announced: Anne Elizabeth Alice Louise. Her official title was Princess Anne of Edinburgh.
She took her middle names from relatives: Elizabeth was after her mother and grandmother, Alice from Princess Alice of Battenberg - the Duke of Edinburgh's mother, Louise was the name of the eldest daughter of King Edward VII - her great-great grandfather.
The Westminster Registrar went to Clarence House to complete the birth certificate.
Are they not sufficiently 'in the know?' 