As the U.S. women’s soccer team was celebrating its fourth World Cup title this summer, lobbyists hired by the U.S. Soccer Federation were in Washington, D.C. arguing that the women’s team, which has filed a gender-discrimination suit against the federation, is fairly compensated.

Using a digital presentation prepared with U.S. Soccer, the lobbyists explained that women get far less in World Cup prize money from FIFA, soccer’s world governing body, in part because the 2019 women’s tournament was projected to generate a tiny...