President Donald Trump is outsourcing much of the response to ex-FBI Director James Comey’s testimony on Thursday to the Republican National Committee, leaning on a team of about 60 staffers to lead a political offensive aimed at Democrats as well as the nation’s former top law enforcement officer.

The White House press team will also refer more substantive questions to the president’s outside counsel, Marc Kasowitz, as they have done since the New York-based attorney’s hiring was announced last week.

The...