WASHINGTON—As the administration’s most ardent senior-level advocate of securing a cease-fire in Gaza, Vice President Kamala Harris’s elevation to the Democratic Party’s standard-bearer now puts her in a stronger position to advocate for a shift in U.S. policy toward Israel and to carry out those changes should she win in November. Though Harris has been careful not to contradict Biden, she has often pushed the envelope of the administration’s messaging on the Israel-Gaza conflict. She has at times advocated forcefully, and ahead of other administration officials, for limiting civilian casualties and addressing the humanitarian crisis in the enclave. “I think...