A Whitney exit permit is not easy to get at that time. The reservations are long gone and there are only 10 walk-ups for any given exit day, which may have all been given out to people who left earlier further up north.
The shortest way to summit without an exit permit is in and out from Horseshoe Meadows, never going down the east side of Whitney. No quota if you do it that way. About 30 miles from trailhead to summit.
