If you disable the built-in delete icon using eventDeleteHandling="Disabled" you will be able to add a custom icon using active areas. See also an example in the docs (https://doc.daypilot.org/scheduler/event-deleting/):
<div id="dp"></div>
<script type="text/javascript">
  var dp = new DayPilot.Scheduler("dp");
  dp.onBeforeEventRender = function(args) {
    args.data.areas = [
      {
        top: 3,
        right: 2,
        bottom: 9,
        width: 17,
        onClick: function(args) { dp.events.remove(args.source); },
        visibility: "Visible",
        css: "event_action_delete"
      }
    ];
  };
  // ...
  dp.init();
</script>
Use visibility: "Visible" to make sure that the icon is always visible.