In an open display of frustration, federal prosecutors on Tuesday night told the judge overseeing former President Donald J. Trump’s classified documents case that a “fundamentally flawed” order she had issued was causing delays and asked her to quickly resolve a critical dispute about one of Mr. Trump’s defenses — leaving them time to appeal if needed.

The unusual and risky move by the prosecutors, contained in a 24-page filing, signaled their mounting impatience with the judge, Aileen M. Cannon, who has allowed the case to become bogged down in a logjam of unresolved issues and curious procedural requests. It was the most directly prosecutors have confronted Judge Cannon’s legal reasoning and unhurried pace, which have called into question whether a trial will take place before the election in November even though both sides say they could be ready for one by summer.

In their filing, prosecutors in the office of the special counsel, Jack Smith, all but begged Judge Cannon to move the case along and make a binding decision about one of Mr. Trump’s most brazen claims: that he cannot be prosecuted for having taken home a trove of national security documents after leaving office because he transformed them into his own personal property under a law known as the Presidential Records Act.

The prosecutors derided that assertion as one “not based on any facts,” adding that it was a “justification that was concocted more than a year after” Mr. Trump left the White House.

“It would be pure fiction,” the prosecutors wrote, “to suggest that highly classified documents created by members of the intelligence community and military and presented to the president of the United States during his term in office were ‘purely private.’”

At a hearing last month in Federal District Court in Fort Pierce, Fla., Judge Cannon herself expressed skepticism about Mr. Trump’s assertion, saying it was most likely not enough to dismiss the case before it went to trial.