For 24 hours last week, Trey Gowdy, the former South Carolina congressman best known for leading congressional investigations of Hillary Clinton, was the new face of President Trump’s outside legal defense and a symbol of a streamlined effort to respond to a fast-moving impeachment inquiry.

A day later, the arrangement fell apart, with lobbying rules prohibiting Mr. Gowdy from starting until January, possibly after the inquiry is over. Now, according to two people familiar with events, Mr. Gowdy is never expected to join the team. And Trump advisers are back to square one, searching for a different lawyer.

How a celebrated announcement quickly ended in disarray offers a rare public glimpse into the internal posturing — and undercutting of colleagues — that has been playing out in the West Wing on a daily basis since Speaker Nancy Pelosi announced a formal impeachment inquiry last month. Even as the White House confronts a deepening threat to Mr. Trump’s presidency, it has struggled to decide how to respond, and who should lead that response.