An effort to inscribe President Trump’s name on the John F. Kennedy Center for the Performing Arts was stalled on Thursday, when a federal judge asked officials to delay the project while he considered a legal objection from a Democratic lawmaker.

The center had said it planned to add the inscription — which would say “renovated and restored by President Donald J. Trump” beneath the building’s main signage — as early as Sept. 8.

Democratic officials have characterized the decision by the center’s Trump-allied board as an aggressive end run around an order by the judge, who said the center had unlawfully renamed the building the Trump-Kennedy Center. In May, Judge Christopher R. Cooper ordered the president’s name removed from the building, citing a 1964 law dedicating the memorial strictly to Kennedy.

At a hearing on Thursday in Federal District Court in Washington, Judge Cooper told Justice Department lawyers that unless they agreed to delay the inscription, he might have to issue a ruling blocking the center from moving forward.

“If the government doesn’t see fit to extend this deadline a bit,” Judge Cooper said, “there may be a bit of a temporary holding pattern while we give these issues the attention they deserve.”

Lawyers for the Justice Department, which is representing the center, did not immediately agree to a delay.

Representative Joyce Beatty, a Democrat from Ohio and an ex officio board member who is suing the center, has objected to the inscription and asked for it to be blocked. Her lawyers argued that it would be in “outright defiance” of Judge Cooper’s order to remove Mr. Trump’s name from the building. The center is appealing that ruling, but in the interim, it has taken the letters down, while devising a workaround that might satisfy the judge.

The board said the inscription was intended to honor the president for his role in securing $257 million from Congress for major renovations to the building.

“I’m worried that they might do this overnight,” Nathaniel Zelinsky, a lawyer for Ms. Beatty, said at the hearing.

Brantley Mayers, a Justice Department lawyer, argued that the inscription would be fully in line with the ruling that the center could not be renamed without the approval of Congress. He said the reference to Mr. Trump amounted to an “acknowledgment” of his role as a “builder and fund-raiser” backing the project.

The board also approved a second inscription on the building that would be added below the first once an endowment fund dedicated to the president reaches $100 million. It would say “endowed by the Trump Kennedy Center Fund.”

“Neither inscription names the center for President Trump or anyone else other than President Kennedy,” Mr. Mayers said.

Justice Department lawyers have argued that the renovation project is desperately needed — and that related fund-raising is contingent upon Mr. Trump being “appropriately” recognized on the building’s facade. They argued that without that recognition, donations would “dry up.” But The Washington Post reported this week, based on internal financial documents, that revenues suffered significantly after Mr. Trump’s name was added to the building in December.

This week, the Justice Department raised the alarm of Democratic lawmakers by suggesting that if the renovations did not proceed as planned, the building would continue to deteriorate and would ultimately have to be demolished.

The center said the suggestion was a “hypothetical scenario” of what could happen if the renovation was not allowed to move forward.

The board voted this month to close the center’s main building for two years to facilitate the renovation, ratifying for the second time a decision Mr. Trump announced in February. Judge Cooper ruled that the board had not sufficiently scrutinized the closure plan in its first vote. Now that the board has voted again, the judge will soon consider whether to allow the closure to move forward.

Ms. Beatty’s lawsuit has argued that the renovations are a cover for a closure necessitated by financial difficulties that have stemmed from Mr. Trump’s takeover. Since he was installed as chairman of the board, the center has seen an exodus of audience members, donors and artists unwilling to associate with an increasingly politicized institution.

But Kennedy Center officials have said that the renovation is an urgent priority, and that keeping the center open during the work would introduce significant cost and safety issues.

Before the hearing, Commerce Secretary Howard Lutnick, who was recently tapped by Mr. Trump to help oversee the center’s affairs, suggested that if the renovation plan were not allowed to proceed, Mr. Trump would walk away from the institution.

“If they don’t agree to let him close and renovate it, because the court steps in, what do you want the guy to do?” Mr. Lutnick said. “He’s only the president for two and a half years.”

Mr. Lutnick said the center would not be immediately demolished if Ms. Beatty prevailed in the lawsuit. But he predicted that if the building were not renovated under Mr. Trump’s plan, it would deteriorate beyond the point of repair.

“It’s just an eventuality,” he said.