Lonnie G. Bunch III has served as a museum director, educator and historian — all positions in which he has demonstrated a skill for diplomacy. But perhaps no amount of charm or discretion can help him avoid the fight he now faces over the direction of the Smithsonian Institution, the cultural behemoth he stewards.

In an executive order last month that accused the Smithsonian of promoting “narratives that portray American and Western values as inherently harmful and oppressive,” President Trump called for an end to spending on exhibitions or programs that “degrade shared American values, divide Americans by race or promote ideologies inconsistent with federal law.”

What that order will mean in practice is not yet clear, given that Mr. Trump does not directly control the institution. But if the White House were to push for significant changes in programming, Mr. Bunch could face stark options — accede to the president’s demands, resign in protest or resist and perhaps be forced out.

Mr. Bunch, the founding director of the National Museum of African American History and Culture, has a track record of working with officials from both sides of the aisle. Sam Brownback, then a Republican Senator from Kansas, sponsored the museum’s founding legislation, which was signed into law by President George W. Bush. Mr. Bunch rose to become the secretary of the Smithsonian in 2019 during Mr. Trump’s first term as president.

Though the men do not share a personal relationship, Mr. Bunch gave Mr. Trump a tour of the museum in 2017, after which the President online called the museum “A great job done by amazing people!”