One can only guess. I did it on October 12, 2001, and conditions were quite nice. Ice at the cables, but no new snow about. Windy on top, and a bit chilly at my start, 5;30 a.m.
It changes from year to year. A matter of luck. There was 2-3' of new snow in the High Sierra, in late September, one year. Follow the reports on this website; they are extremely informative. Get the Lone Pine forecast right before you leave. There's a live cam on sometimes.
Good luck!