Mike Pence used to dream of following two successful Donald Trump terms with his own turn in the White House. Now, sitting in offices with a great view of the National Archives and adorned by photos of Republican presidents, Pence is trying to find a way for his brand of conservatism to outlast Trump’s. To do that, he’s steered his think tank, Advancing American Freedom, into a battle with the Heritage Foundation, a citadel of American conservatism since the days of Ronald Reagan. Heritage has shifted in a distinctively pro-Trump direction — echoing the party Trump has remade in a...