I may have found the answer: http://doc.daypilot.org/month/angularjs/
It seems the DayPilot control can be accessed from the $scope:
$scope.message = function() {
$scope.dp.message("Hi");
};
Given that, you should be able to set the startDate (or any property for that matter) and then call $scope.dp.update().
BTW: I believe the "dp" property off the $scope object is derived from the id attribute used in the DayPilot directive in the HTML.