WASHINGTON—The Library of Congress—long a sleepy, noncontroversial staple of the nation’s capital—has become the latest battleground in President Trump’s power struggle between the branches of government.

Lawmakers are raising separation-of-powers concerns as the White House seeks to install a Justice Department official at the legislative-branch institution.

Library of Congress employees on Monday turned away two Trump appointees seeking to take over the Copyright Office. "For the time being, the acting librarian is the acting librarian," Sen. Alex Padilla (Calif.), the top Rules Committee Democrat, said Tuesday, and Trump's people "seem to be respecting that."