Steve&Julius- May 15th is opening day for the N Rim- so the trails and especially the N Rim complex are going to be very, very busy. There will be alot of company on the trails.....
Steve- There is a Hiker shuttle that leaves the Backcountry Office and Bright Angel Lodge at 5,6 and 7am.; and is a direct ride to the S. Kaibab Trailhead. The advantage of this is that Julius (and his 7 yr old daughter) coming out at the BA Trailhead will be closer to his car.(Walking or 2 shuttle stops) The 5 am shuttle puts you at the S. Kaibab Trailhead at 5:30, the river around 9:30 and through "the Box" on the N. Kaibab by 11am. This would be the latest I'ld start; as it puts you on the 5 mile, no shade portion of the N. Kaibab (and the hottest section)in the middle of the day. Good news its relatively flat, so one can make good time from Phantom Ranch/BA Campground to Roaring Springs in under 4-5 hrs. The accumilation of the altitude, heat and desert climate will start taking its toll on the Roaring Springs to Trailhead section- 4000' in 4 miles at 4-8k elevation.
If you want to leave earlier, the two options are: parking at the BA trailhead/Backcountry Office and taking a taxi; or parking at the Parking lot at the picnic area at the road junction to the S. Kaibab/Yaki Point; and adding another 1/2 to 3/4 mile to your walk; and making it a little more challenging to Julius to get his car.
If Julius has to go to this parking lot- he'll take 2 shuttles and over an hour to get to the S Kaibab Trailhead stop, and walk back to his car; provided he is out of the canyon by sunset. Otherwise he'ld need a taxi as well. Julius's daughter will determine how soon they 'top out' (but should plan a late afternoon stop at the ice ream store near the BA trailhead.)

Camping at S Rim that weekend will be challenging. Mather Campground will be full(Check Rec.gov); there may be a chance of a cancelation and get a site as a walk-up. The Forest Service Campground TenX , outside the park; probably will have space- just a little farther from the S. Kaibab Trailhead. There is National Forest just outside the S. Rim Entrance- so 1/4 mile off the road and its the cheapest option.
It would be the same on the N. Rim- "no room in the inn or campground" if you don't have a reservation by now. There is Forest Service land outside the Park near Jacobs Lake Lodge that provides that free camping to rest after your 21 mile walk.
The S. Kaibab vs Bright Angel going north: the elevation change and distance are very close from rim to the river- less than 50 m in elevation. The difference is the 2 mile River Trail walk on the BA to the N. Kaibab; and the S. Kaibab is a direct cross of the river and to Phantom Ranch/BA Campground complex. The BA has water stations at 1 1/2, 3 mile and Indian garden (half way),vs carrying 3-4 liters on the S Kaibab.
One should always have a way to treat water on the N. Kaibab; because: if the water lines are broken, there will be no potable water on the N. Kaibab. The Good news is that the trail parallels the BA creek to Roaring Springs, so the worse case would be needing 4 liters at Roaring Springs to get to the rim if the water at Supai Tunnel is off. There have been the typical line breaks this winter(the line currently has a break between the Box and Ribbon Falls.) The Back Country Office updates the water status on the park website, and you can check once at the park for last minute planning/update.