> A friend and I are hoping beyond hope...
The day before you hike, be at any one of the Inyo National Forest ranger stations (Lee Vining, Mammoth, Bishop, Lone Pine) at 11 AM, when the walk-in permits become available. 40% of those permits are available on a walk-in basis, so there should be 4 available at that time. I think your chances should be pretty high.
Not sure when "no-show" permits for those overnight permits become available -- maybe the day of your hike (I'd call the permit office and ask about them), but chances are good that you might get those, too.
Just pick an alternate trip to use for plan B, but I don't think you will need it.