Figured it out!
First I passed what Iceable suggested and also added moment script above, like so
wp_register_script('moment', "https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.22.2/moment.min.js");
wp_register_script( 'fullcalendar', "https://cdnjs.cloudflare.com/ajax/libs/fullcalendar/3.9.0/fullcalendar.min.js", array( 'jquery' ), null, true);
and then I edited JS like so
jQuery(function ($) {
$('#calendar').fullCalendar({
})
});
And it showed up!