Since Sam Bankman-Fried was convicted of fraud last year, he has hired a new lawyer known for courtroom showmanship. A group of sympathetic law professors has pushed for a reappraisal of his actions. And his parents have turned for help to former employees of FTX, the collapsed cryptocurrency exchange he founded.

From a federal detention center in Brooklyn, Mr. Bankman-Fried, 31, has continued to fight his case behind the scenes, as he aims for a lenient sentence and prepares to appeal his conviction. On Tuesday, his lawyers filed a legal memo in U.S. District Court in Manhattan, arguing that he should receive a prison sentence of between five and a quarter and six and a half years.

Mr. Bankman-Fried is “deeply, deeply sorry” for “the pain he caused over the last two years,” the memo said. “His sole focus after the collapse of FTX was making customers whole.”

The filing was a crucial step before Mr. Bankman-Fried’s sentencing on March 28, when the federal judge overseeing his case, Lewis A. Kaplan, will decide how long to imprison the onetime billionaire on charges that carry a maximum sentence of 110 years. But it was only one prong of a long-shot strategy orchestrated by Mr. Bankman-Fried’s family and friends to reverse his conviction and engineer a public reappraisal of his leadership at FTX.

Since last year’s trial, Mr. Bankman-Fried has hired Marc Mukasey, who once represented former President Donald J. Trump, to oversee his sentencing, as well as a separate lawyer at the law firm Shapiro Arato Bach to handle the appeal. His parents, the Stanford University law professors Joe Bankman and Barbara Fried, have also been involved in the defense, helping line up people to write letters vouching for their son’s character that were included in the sentencing memo.

In an interview, Natalie Tien, a former assistant to Mr. Bankman-Fried at FTX, said she had written a letter for the memo after exchanging emails with Mr. Bankman and Ms. Fried.