That's a tough call with 6 different people. If you press ahead and plan this, and get to this point, be sure you have a Plan B alternate trail in mind.
I'd suggest plans A, B, and C might be:
A: camp at Horseshoe Meadows on a Saturday in early September. On Sunday, drive down to the Visitor Center by 11 AM and try for a Monday entry date for 6.
B: if unsuccessful, get back up to HM and day hike. On Monday, drive down and try for a Tuesday entry date.
C: if unsuccessful, repeat for a Wednesday date.
I'd lay 99.9% odds you will get a permit one of those days.
Check the
2009 unused permit numbers. Even Labor Day had a number of unused overnights available.