This is thought to be partly due to the fact that the film is going through last minute edits to include details of the Duchess of Sussex's apology for misleading a court over whether she gave authorisation for an aide to brief the authors of a biography about her.
But the Palace is said to have threatened to refuse co-operate with the corporation on future projects if they are not given the right to respond to the full documentary, which airs on Sunday night on BBC Two.
The BBC is also said to be sensitive about its relationship with the royals in the wake of the Martin Bashir scandal. As a result, it is understood they are going through The Princes and The Press line by line, to ensure sensitivity.
Yet despite a series of meetings between the Duke of Cambridge's aides and the BBC, they are determined that the "investigation" will not be shown to courtiers.
It is unclear whether the Duke and Duchess of Sussex have raised any concerns about the show or were contacted before the other households complained.
A senior royal source told the Mail on Sunday that the Queen was "upset" that the "tittle-tattle" documentary would air without anyone at the Palace having a chance to see it.
It is the second time the Royals have intervened over coverage of the rumoured feud between the Duke of Cambridge and his brother, the Duke of Sussex.
Earlier this year, just hours before the documentary Harry And William: What Went Wrong? was due to broadcast, ITV removed claims by Omid Scobie, the Duke and Duchess of Sussex's biographer, that Prince William and his staff planted a story about Prince Harry's mental health.
There is concern that the BBC documentary may repeat similar allegations, which have been consistently denied by the brothers.
The BBC said the show will examine the years in which the Duke and Duchess of Cambridge and the Duke and Duchess of Sussex have charted "very different courses" in their relationships with the media.
The first of the two-part documentary covers the "years leading up to and including the engagement and marriage of the Duke and Duchess of Sussex" and provides "context" for their relationship with the press, by examining the "illegal activities" of some news outlets in the 1990s.
Rajan, who is tipped to be a frontrunner to succeed Laura Kuenssberg as the BBC's political editor, has been a vocal critic of the monarchy.
In a 2012 column for The Independent critiquing the media's relationship with the royals, he described the notion of a hereditary monarchy as "absurd" and Prince Charles as "scientifically illiterate".
Any boycott could put under threat joint projects between the Royal Family and the BBC, which have recently included a tribute to Prince Philip and the documentary series Earthshot which was presented by the Duke of Cambridge.
The BBC also films the Queen's Christmas speech as part of a rota with ITV and Sky.
On Sunday, The Palace refused to comment on the documentary or any complaint to the BBC.
A BBC spokesman said: "The programme is about how royal journalism is done and features a range of journalists from broadcast and the newspaper industry."
