I second camping at Horseshoe Meadows on the night of the 27th. It would probably even be a good idea to spend some time walking around a little in the area on the 28th in order to better acclimate. Donâ€™t do too much, as you donâ€™t want to exhaust yourself for your climb up Whitney. The key is to spend some time up at around 10000+ feet to get acclimated. Then, on the night of the 28th, stay at Whitney Portal so that you will be able to get right to things when you get up. The altitude here is not as high as Horseshoe Meadows (around 8000ft). But it is still decently high to get you SOME acclimation.
On the other hand, staying in Lone Pine before your hike is NOT the greatest idea. The elevation here is only 4000ft, which is not going to acclimate you for anything. If you wish to stay in a motel room in Lone Pine, save it for after your hike.