G4u2, when you start your trip setup, on this screen:https://www.recreation.gov/permits/233262/registration/detailed-availability
in the upper left, it shows Overnight. Click that, and change it to "Overnight exiting Mt Whitney". Then set your date and group size, then "Is this a commercial guided trip" = No.
That should open up the list of starting points where you scroll down to Cottonwood Lakes. Select your entry date.
THEN, note there are TWO tabs below the Yes/No question. Click the "Exit From Mt. Whitney" tab, and there you get to select your exit date.
edit: just for fun, I stepped through the "Questionnaire" version using my Firefox version, and selecting "Yes, ending at Mt. Whitney" (which really means "exiting via Trail Crest"), and it stops at:
Finished, You will need a... Overnight Exiting Mt Whitney Permit
but the button to "Check Availability" is disabled. USE Google Chrome, NOT Firefox!