Hedge-fund manager William Browder has long portrayed himself as a crusading financier seeking justice for the death of his associate Sergei Magnitsky, who was allegedly beaten to death in a Russian prison in 2009 after exposing a tax fraud involving government officials there.

According to a Russian businessman accused in Manhattan federal court of laundering some of the proceeds of that alleged $230 million fraud, Mr. Browder’s account is a lie.

Lawyers...