WASHINGTON — With the Federal Reserve under more intense attack than at any time in decades, Ben S. Bernanke, the professorial chairman of the central bank, was schooled last month in how to handle the increased political demands of his job.

Representative Barney Frank, left, said of Ben S. Bernanke, the Fed chairman, right: “Ben Bernanke turns out to have better political instincts than anybody thought.”

For months, he had warned — without anyone on Capitol Hill appearing to listen — that a seemingly innocuous bill to let Congress “audit” the Fed would gravely threaten the central bank’s independence.

It was alarming enough that the bill’s author was Representative Ron Paul, the...