Just weeks after President Trump took office for a second time, one of his most prominent supporters, Rudolph W. Giuliani, was 4,500 miles away wearing a different hat.

Instead of the red “Make America Great Again” ball caps that have been a defining feature at Trump rallies for the last decade, Mr. Giuliani’s new hat promised to “Make Srpska Great Again.”

His appearance in the hat at a political rally for Milorad Dodik, the president of the small self-governing territory in Bosnia and Herzegovina, marked the public debut of a costly and wide-reaching influence campaign aimed at persuading the Trump administration to come to the defense of a pro-Russian authoritarian trying to fend off threats to his reign at home and in Washington.

The effort has grown to include well-connected lobbyists and lawyers with ties to Mr. Trump who are being paid about $300,000 per month or more, according to filings with the Justice Department.

Their goal is to present Mr. Dodik, a towering figure in Republika Srpska, as a sort of Trump of the Balkans.

The campaign has cast Mr. Dodik, who is contesting an effort to remove him from the presidency and fighting to lift U.S. sanctions, as a victim of the same sort of political prosecutions that Mr. Giuliani and Mr. Trump claim they have faced, and as someone willing to cut lucrative deals with Washington.