Our current plan is to start at the South rim, take Bright angel down and then head up to the North rim via the Kaibab trail (unless we end up getting a permit to camp on the bottom

I have a few questions;
What are the chances of getting a walk up permit? Not expecting to get one but it would be awesome to camp on the bottom of the canyon!
Here is a list of what I am bringing, should I bring more? (Hammock; instead of a tent, sleeping bag, 1st aid/survival kit, 100 oz camelback, 20 oz additional water bottle, 20 oz electrolyte drink, knife, flash lite/ headlamp, water purifier, miniature stove, food). I feel like I have enough stuff but it would be nice to hear from a more experienced hiker.
Any comments, suggestions, tips or anything would be appreciated. I am really excited about this and I hope everything goes well!