Starting your overnight hike on a Sunday, your chances are very good. Saturday less. Go in on Saturday AM and ask if there are any available (from other groups picking up permits and shedding some slots). On Sunday at 11 AM any no-shows will become available for the Sunday start. That late in the year, there may be some slots available for a Saturday start still left Saturday.
Last year's unused permits show an anomalous low number left for the last weekend in Sept. Not sure why that occurred -- maybe the earlier weeks' closed trail.
Plan B: Extend your trip an extra day. I am 99.9% sure you would get permits to start on Monday.
Plan C: Unlikely you will need it. And coming from so far, I can understand your intense wishes for Whitney. But Mt Langley via Cottonwood Lakes would give you an excellent experience above 14k. Or drive north and dayhike White Mountain on Monday or Tuesday.
