I think all the campgrounds will be packed on Labor Day weekend. However, if you sleep in your car, and don't draw attention to it (arrive late, etc), trail heads parking lots can be used. There is short section of old road just east of the Tioga Pass gate that people use sometimes -- you could drive there Friday night and sleep in the car.
I would sleep as high as possible Saturday night. End of the road at Rock Creek, just S of Mammoth is at 10k.
Sunday night, you could use walk-in sites at Whitney Portal, (but you might want a tent for that). ....but there is nothing wrong with throwing out a pad and sleeping bag without a tent. Makes packing up before you start hiking easier. That would be quicker (and better for acclimatization) than driving up from Lone Pine. (Sleeping in the car is an option, but I'd try to get a spot in the backpacker parking lot. Wouldn't want to be in the car along the road, with other cars driving by making noise every 15 minutes.)
When you are getting ready to start from Whitney Portal (when it is dark), beware that bears may be watching, and if you put your day pack down anywhere but inside the car or a locker (with doors closed!) a bear can dash in, grab the pack and run. It happens all the time.
