I don't know what your available time slots are, but I presume you've
looked at all the available permits around other trails (Inyo National forest) on the recreation.gov site. I believe some allow you to exit Mt. Whitney. FYI, I never had a problem with getting a walkin permit at the Lodgepole Visitor Center for Sequoia/Kings Canyon trails. You could come after the quota season on Whitney, but be prepared for snow.
Best Regards,
Rick