Thanks. The Trail Crest Exit permit you speak of is simply obtained from either Lone Pine or Bishop Ranger office and has nothing to do with lottery? Right?
The required permits are outside of the lottery. You need both a Kearsarge Pass entry permit AND the Trail Crest Exit permit. They are made available online starting 6 months before your hiking date, and there is a quota, which fills up pretty fast. In fact,
checking online, nearly all exit permits are already gone for July 13 - Aug 22. Kearsarge Pass permits seem to be more readily available.
In your favor, though, is the fact that 15 are reservable, and 10 are held back for walk-ins. So make that entry reservation NOW, and then when you go in to pick up the permit, ask for a trail crest exit permit.
Steps on reserving your wilderness permit are outlined
in Box 3 of the Wilderness Permit Options page (link is above on the left). You would choose the "Overnight Exiting Mt Whitney", Trail is "Kearsarge Pass JM31".
Enter any date close to your hike date, and it will then let you browse forward and back through dates, 2 weeks at a time. And since you chose "Overnight Exiting Mt Whitney", it will also give you the dates and available numbers on the exit permits on the same screen.
Good luck, it is a bit confusing at first.
Once you have made the reservation online, you can then pick up your permit in any Inyo N.F. ranger station: Lee Vining to Lone Pine and in between. The permits are available up to 2 days in advance of your hike date.