I visited the Grand Canyon in late May 2011 during a full moon. If you are planning to shoot the full moon rising or setting, you'll almost certainly need to do so from the north rim. The angels are wrong on the South Rim. I had a trip scheduled far in advance and was not able to rearrange for the best angles for the moon.
Laurent Martres books to the southwest are the best resource on location specifics. The Grand Canyon is a huge park. The locations on the rim are popular and crowded. You'll need to use bus service on the south rim.
Moonrise is typically better a day before the full moon, and moonset better the day after. I've seen some great images from the bottom of the canyon lit just by moonlight. Even a hike part way down would be better than trying to photograph from the rim.
The Photogrpahers Ephemeris is a great software program to look at angles for moonrise and moonset.