You gave some alarm clock applications as examples, so I think you can start from understanding what this alarm clock icon in status bar is, and how it is shown there.
I note, that, as you can see, application "Alarm Clock Plus" doesn't set its own icon to status bar, just system icon for alarm.
You can see here how you can manipulate this icon on Pre-Lollipop only:
protected void setStatusBarIcon(boolean enabled)
{
Intent alarmChanged = new Intent("android.intent.action.ALARM_CHANGED");
alarmChanged.putExtra("alarmSet", enabled);
sendBroadcast(alarmChanged);
}
If you can give an example of application that puts its own (different from system alarm icon) icon to status bar (not as notification), it could help to see that it is possible.
Unfortunately, I don't think that it is possible.
Even just logically: there are two parts of status bar – user (notifications) and system (other). If system allows to put something to system part – there will be so much useless icons (posted by other applications), and there will be no consistency, and it might break all of the use experience. I think.
Hope it helps