I presume that you want to stay @ Bright Angel Camp? One option would be to call (1-5PM) and ask if there are any dates that ARE open. And, if so, immediately fax in a permit request for THAT date.
Second, you could try getting a permit for Cottonwood and backpack to Cottonwood on day 1and take the next day to go spend the day @ Phantom, etc. and then stay @ Cottonwood again. That way you'd at least have ONE option if the walk-up doesn't come in.
The North Rim, as you likely know, is a he@@ of a lot less crowded that the South Rim, BUT, they both issue permits from the same pool.