In a Congress controlled entirely by Republicans, there has been little pushback to President Trump’s brute-force efforts to unilaterally upend entire federal agencies and bend them to his will. But the administration’s latest takeover target is much different from others that have faced Trump loyalists at their front doors, trying to assert control.

The Library of Congress, with its grand Beaux-Arts architecture and iconic reading room, is a distinctly congressional institution just across First Street from the Capitol. It is part of the history of the Capitol itself, as British soldiers used books and maps from the original library inside the unfinished building to stoke the flames when they torched the Capitol in August 1814. Thomas Jefferson then sold Congress his own books to restart the collection.

Members of Congress are proprietary about the library, a sentiment that has provoked bipartisan resistance after Mr. Trump summarily fired the popular chief librarian and tried to install one of his lawyers as the new acting head of the institution that catalogs American literature and culture.

“We’ve made it clear that there needs to be a consultation around this,” Senator John Thune, Republican of South Dakota and the majority leader, said this week. He suggested that the White House had overstepped its authority and that both Congress and the president play roles in deciding who leads the library.

Other Republicans have joined outraged Democrats in signaling their discomfort with the president’s meddling in a squarely congressional institution after he abruptly fired Carla Hayden as the librarian of Congress. Mr. Trump then moved to put Todd Blanche, his former personal lawyer who is now at the Justice Department, in her place. That prompted a brief standoff at the library this week and a rebellion of sorts among top staff members there, who argue they answer to Congress, not the White House.

The episode has given rise to a quiet battle over the separation of powers centered on a relatively obscure corner of the government. The outcome could determine not only the leadership of the library and its vast collection, but also whether members of Congress can continue to receive nonpartisan research materials on a confidential basis and who controls the immense repository of copyrightable material in the United States.