From the 12th floor of a glass office tower in the Washington suburbs, a campaign to sway the governor’s race in Kentucky on Tuesday is being waged with an alarmist claim that has little to do with the race itself: If Democrats have their way, soon boys will be able to compete against girls in school sports. This scenario, presented in a pair of ads that are appearing on computer screens and smartphones across Kentucky, is the work of a little-known group funded by anonymous donors called the American Principles Project, which in recent years has focused on fighting more familiar clashes in the culture wars over same-sex marriage and abortion rights. The group is limiting its work to Kentucky for now, but strategists say it has bigger ambitions.