No backcountry permit is required if you hike Rim-to-Rim in one day: I did this last month, hiking the Kaibab from the S. Rim to the N. Rim late that day, then hiked in to the Hiker-Biker area of the N. Rim campground. If you hike or bike to this spot ($6/night, or half that if you have a Sr. Park Pass) you need no reservation.
(I did the reverse of this on the return, 'tho I didn't camp on the S. Rim....I suspect there's a hiker-biker area in 1 of the S. Rim CG's w/the same policy).
Happy Trls., all
Jon Phipps/Rodknee Dangertrl.