I haven't done this. But if I was going to do this, I might camp at Lone Pine Lake instead of Outpost Camp on Day 1. Lone Pine Lake is beautiful, and at night it is quieter (you don't have people walking through the camp all night on their way to the summit.) During the day, it can be a bit busy, as dayhikers go to LPL to fish.
The first two times I hiked to the summit of Whitney, I stayed at Trail Camp both nights, and summitted in between night 1 and night 2. That worked for me. Perhaps an alternative itinerary to consider.