The school is also an unofficial feeder to the Delphian School in Oregon, a $42,000-a-year boarding school that counts Cruise and his ex-wife Nicole Kidman's adopted children Connor and Isabella, 17 and 19, among its alumni.
Almost half of Delphian pupils are members of the church.
"[Suri] is coming to an age where she gets educated enough to get locked into the faith," Marty Rathbun, a former senior executive in Scientology told the Independent. "That's why there's almost certainly truth in the consideration that schools have started coming into play in all this."
Cruise is Scientology's most celebrated member. He and Holmes were married in 2006, at a ceremony presided over by the church's leader, David Miscavige. But they are said to have had growing disagreements over how the faith should be applied to rearing their child.
"Scientologists believe in reincarnation, that infants are essentially ancient adults being in children's bodies," a source familiar with the dispute said.
"Tom treats Suri as a little adult. Katie takes the opposite view: that she deserves a childhood."
Importantly, Suri has reached the age at which church members begin being "audited", a form of counselling at the centre of Scientology.
A "security check" devised by Hubbard for 6 to 12-year-olds requires them to be asked personal questions while attached to an "e-meter", measuring electrical charges carried by their body.
The list of questions includes: "have you ever got yourself dirty on purpose?" And: "have you ever told bad stories about someone?"
Holmes is apparently anxious to prevent her child from being subjected to any such "check". She is therefore reported to be seeking sole custody of Suri, a move that would give her the right to determine her religious upbringing and where she is educated.
The actress filed for divorce in New York, while Cruise, 49, has counter-filed in Los Angeles.
Both states will take a similar view regarding the distribution of the couple's $275 million fortune, but New York courts tend to grant sole custody, while joint custody arrangements are more common in California.
Holmes is perhaps also anxious to prevent history repeating itself. When Cruise divorced Kidman, he secured a joint-custody agreement. The couple's children grew up as members of the church and today have a distant relationship with their mother, who is not a Scientologist.
Rathbun says he "audited" Cruise at Scientology's headquarters in Los Angeles between 2001 and 2003, when the actor was divorcing Kidman, and claims to have witnessed this process. He said he saw Connor and Isabella, who were 6 and 8 at the time, being introduced to auditing.
"Everyone was thrilled to see Connor playing on an e-meter," he said.
The church did not respond to queries about that yesterday. But it has attacked Rathbun's credibility, describing him as a "liar", a "criminal" and an "apostate".
Elsewhere, it was reported that Holmes was anxious to stave off an imminent effort to enrol Suri in the Sea Org, the church's clergy, which requires recruits to sign a billion-year contract. But expert sources pointed out that the Sea Org does not usually welcome new members until they are teenagers.
Rathbun said, however, that schools using "study technology" - an educational technique involving rigid study programmes, repeated use of dictionaries and a strong emphasis on building clay models - can provide a long-term path to membership in the organisation.
"In the old days these schools would use study tech as a small part of a traditional curriculum," he said. "Now things have evolved, to the point where you hear of pupils are being assigned 'lower ethics conditions' and told to do auditing and getting an education which revolves around how to be 'good' Scientologists. Many of them are becoming a recruiting ground for the Sea Org."
The church has yet to comment on its role in the divorce, except to deny extravagant reports carried by the British tabloids that yesterday suggested that it had hired private investigators to follow Holmes around New York.
Behind the scenes, a damage-limitation exercise is taking place. But it is hampered by the absence of Jessica Davis, a church employee who has been its leading point of contact with Holmes for many years. The wife of Tommy Davis, Scientology's longstanding spokesman, she is incapacitated by illness.
As a result, a church source said, "David Miscavige lost his lines of communication with Holmes some time ago".Independent
