Months before the nation’s highest court took up an Illinois dispute over nonmember union fees, opponents of the idea were already fighting the battle.For four days in July, conservative think tanks and attorneys general blanketed the U.S. Supreme Court with arguments in favor of taking up the case in order to strike down the fees.Groups such as the Cato Institute, the Center for Constitutional Jurisprudence and the Pacific Legal Foundation are a few of the advocacy groups that filed amicus briefs in the case, Mark …