However, you don't need an exit permit if you don't want to climb Mt. Whitney. I've heard though, if you want to climb Mt. Whitney, a permit is much easier to get if you enter and leave from the west.
You might hike from Horseshoe Meadows over Trail Pass, down to Corral Meadow to the Kern River. Then you hike north up the river and head east at Wallace Creek or Tyndall Creek. Wright Lakes and Wallace Lakes basins are supposed to be spectacular and you can bag peaks out of both.
You can follow the river to its headwaters in the Lake South America area - an incredible basin of lakes at 12,000'. There is fishing all along the Kern River. You have a couple choices to cross over to the PCT from the headwall at the south end of the Lake South America plateau. When I did this trip four years ago I had an itinerary of campsites lined out at bear boxes. We were NOT required to carry bear cannisters.
Here's a topo of the north end of the trip.
You can also head up into Miter Basin at the head of Rock Creek and the Siberian Pass and Army Pass areas and spend more time fishing and ridge/peak bagging. Supposedly the view over into Whitney Basin is spectacular. I think you could figure out a way to spend seven days in this area with no problem.
Have fun!
Jeff.