There are camp sites just over Tioga Pass at several lakes right on the highway, and also north towards Saddlebag Lake. There should be space mid-week in September, but not sure.
As for the summit climb, watch the weather:
Mt Whitney Weather The NOAA Whitney summit weather link shows highs of 36 - 39 all week, with a chance of snow! With those temps, you definitely need to take extra layers of clothing.
I personally skip the water filter, but most carry one. And carry very little water until you reach Trail Camp.
