Just an idea.
Why not pick a random week. Say week 2 of this year.
I'm going to assume you use the agendaWeek view. If not you could come up with a simulair solution for any of the other views.
This runs from;
1/5/2015 to 9/5/2015 (assuming a 5 day event) so weekends=false
Do not show the header so header=null
You now fill you're events into the calendar in the events callback function.
You come up with start en end times inside you're random week.
So an event on day1 could have a
event.title="First event on day1";
event.start=moment('1/5/2015 8:00', 'MM/DD/YYYY HH:mm');
event.end=moment('1/5/2015 11:00', 'MM/DD/YYYY HH:mm');
The only issue now is that the headers of the day's represent dates. This could be fixed easily by;
$(".fc-day-header.fc-mon").html("Day 1");
$(".fc-day-header.fc-tue").html("Day 2");