I backpacked R2R north to south coming up the Bright Angel Trail. There is some parking right at the Bright Angel trail head. This could be a factor in planning. Us old folks camped two nights, first night at Cottonwood second at Indian Garden. Easy peasy. We luckily had a friend who helped with the logistics. We stayed before the hike in Tusayan at the Red Feather Lodge, left a car at the Bright Angel trail head, drove to the North Rim, and hiked down to Cottonwood to camp the first night.(We were on the trail around 1pm) Our friend hike down with us and camped the first night, hiked back up the North Kaibab Trail, camped at the North Rim, then drove to the South Rim to meet us (he and one in our party then went up to Great Basin Park)
I tried unsuccessfully to get camping permits twice. After some input from a ranger I was successful the third time. There are only 10 campsites at Cottonwood so they are always booked. We went the first week in October and the weather was very comfortable - mid 50's at campsites at night, around 85 mid day at Phantom Ranch.
The Bright Angel Trail is very popular and has two rest area/water stops between Indian Garden and the South Rim. Very conveniently spaced. Lots of tourists come down the trail a ways then head back up.
