I can't tell you anything concrete about that specific wilderness permit office, although I have dealt with it a few dozen times in the past. Many permit offices start issuing the walk-up permits the day before the start day, typically around 10-11 a.m. If you are shooting for the Bishop Pass Trail, either you get there early the day before, or you seriously take your chances on getting zero. I have shown up on the start day to get a permit, but I was standing first in line from pre-dawn hours in order to get what was left from the day before. Backpackers have been trying to "game" that permit system for years and years.
As for alternative entry points, it kind of depends on where you intend to finish. There is Paiute Pass. You could start at North Lake and go over Lamarck Col. You could start at Sabrina and go over Echo Col (class 3, I believe). You can shift your start day by a day. Shifting away from a weekend start is to your advantage.