I have slept in my car at Horseshoe Meadows (Cottonwood Pass trailhead campground) parking area -- next to the walk-in camp sites. I don't think there are any rules against it. I think you could sleep in your car in the Whitney Portal overflow parking lot (the last spur before the trailhead), too -- you want to be as far from the trail head as possible, since hikers start really early and their noise would make for a bad night. No permits required to park.

If you would rather sleep on the ground, the walk-in sites at Whitney Portal are reasonably close to the parking. They are one-night only, first come first served, and cost $5 or $10. But the car sounds more comfortable and less setup hassle. You definitely want to sleep in your car at Whitney Portal Aug 2, so you can get out and start early on Aug 3. Good luck, by the way.

You are wise to sleep at the high elevations. In fact, you could do that every night, and use the showers at Whitney Portal Hostel (In Lone Pine). If you slept as high as you are comfortable every night, you should be well prepared for the altitude on your hike day.