Wow can’t believe I missed this one. Line 108 of ui.datepicker.js:
/* Class name added to elements to indicate already configured with a date picker. */
markerClassName: 'hasDatepicker',
So I just need to test for hasClass('hasDatepicker'). This seems like the most straightforward way. Furthermore, this statement checks whether the datepicker is currently opened (for anyone who is interested):
if ($("#ui-datepicker-div").is(":visible") && $("#ui-datepicker-div").html() != "") {
// datepicker is open. you need the second condition because it starts off as visible but empty
}