Don’t feel alone at not getting a permit. Grand Canyon received over 1000 permit requests for May on the first day of January. Unfortunately, most of the requests involved corridor trail campgrounds so requests were chosen at random from all the requests submitted.
The North Rim opens on May 15. If you are crossing before that date it will probably be easy to get a site at Cottonwood Camp (half-way up to the North Rim). If you are after that date, it would be more difficult because you are competing with lots of rim-to-rim hikers.
Do not attempt to stealth camp in the Grand Canyon, the chance of meeting a ranger on patrol is great and the fine is at least $150. You can not join someone and share their site once you get to the bottom. That would mean too many people on their permit and you both would be in trouble - rangers check permits at least once a day.
A few sites at Indian Garden (half-way down from the South Rim) and Bright Angel Campground (at the bottom adjacent to Phantom Ranch) and Cottonwood Camp (half way up to the North Rim) are always set aside for walk-ins. However, there is a system for getting those spots. You go to the Backcountry Office as soon as you arrive and are given a number. The next morning at precisely 8:00AM, numbers are called and those with low numbers are given the available spots (usually for the following day). Then the numbers are re-issued so you will get a lower number. You then show up at precisely 8:00AM the next morning and it goes on as you move up. You almost always will get a spot by the third day.
Other choices - be sure you tell the Backcountry Office ranger that you are an AZT thru-hiker. They do try to accomodate thru-hikers if possible. Also, be sure you tell them you are able and willing to camp outside of the corridor trails if it does not mean walking too much farther. Here are some options:
Camp at Horn Creek - 1.5 miles west of Indian Garden on the Bright Angel Trail
Camp in the Clear Creek use area - 2.5 miles east of Phantom Ranch
Camp in Cremation Canyon - 1.5 miles east of the South Kaibab Trail
The Backcountry ranger may think of some other options.
Yes, you can cross in one day, but I would try to get a permit or at least talk with a Backcountry Ranger first to see if there are other options. If you cross in one day you will want to start very early in the morning.
Turtle Walking