When are you trying to go?
Here's the link to the Rec.gov page with the Bishop Pass entry and Trail Crest exit:
LinkUnfortunately, the Trail Crest Exit permit is extremely difficult to get, as only 15 are reservable, and they become available at 7 AM Pacific time 6 months before the date you want to exit. The available permits are snapped up on the website in just a few seconds!
If you would exit on ANY other trail, you aren't constrained with an exit permit date, so the permits are much easier to get. You might also try entering a little to the south at Horseshoe Meadows (permit is Cottonwood Pass or Cottonwood Lakes) and exiting at Bishop Pass. That permit should be way easier to get. Starting there adds 1 long day to your trip, and you can still climb Whitney from the west side.
I see the two Cottonwood entries are pretty well booked full now, too. Trail Pass has no quota, and adds only 2 miles to the Cottonwood Pass entry.