“Here’s our fundamental problem: We don’t have an elevator pitch to easily describe this to the right,” Mr. Miller said. He likened the explanations from the White House and its surrogates to the way someone solves the game “Clue,” with its menagerie of possible culprits, weapons and crime scenes.

The show, which is airing on a half dozen stations in Virginia and Florida and streams on the website WarRoom.org, is a remarkably blunt attempt by former Trump aides to criticize and influence the work of current Trump administration officials. Even the show’s name is something of an affront to the White House — and to Mr. Trump himself. He has told aides that he does not want to create a war room, as crisis response operations are often nicknamed, because he is concerned that people would see it as an acknowledgment that he views the impeachment inquiry as legitimate.

Though many in the White House may be tempted to dismiss the show as a bootleg operation run by people cut off from Mr. Trump’s inner circle, Mr. Miller said in an interview that the president ultimately cares about one thing when it comes to managing a crisis of this magnitude: “Are there people defending me or not?”

Mr. Miller added that with the exception of a handful of Trump loyalists, “There is a serious lack of Trump allies jumping out there and defending the president.” Too few Republicans, he said, are questioning the process of the House Democrats’ impeachment inquiry, which he said was being run in a clandestine fashion from “Schiff’s secret Capitol basement bunker.” It was a reference to the secure hearing room in the Capitol where Representative Adam Schiff of California, the House Intelligence Committee chairman, is presiding over witness interviews.

The idea for the new program came to Mr. Bannon after he appeared last week on the radio show of John Fredericks, a conservative whose daily broadcast airs across Virginia. He was floored by the backlash he got when he stated what he said he assumed was accepted as fact among Republicans and Democrats alike: Mr. Trump is likely to be impeached.