Welcome, Camminatore!
Regarding the midnight start: I have been told by National Forest staff that you can hike up to Lone Pine Lake without a permit. It is only beyond the lake, and the Wilderness boundary, that you must have a valid permit. So I suppose you could start at 11 p.m.
But that would be foolish -- the sleep deprivation will catch up to you before you reach the summit. And the summit is only half way! Lots more people leave at 3 or 4 a.m. You won't be alone.
All the info about acclimatization is on the mark. Here are two recent discussions worth reading:
Anyone know about a High altitue B&B or Campground? recomendations for an acclimatization route up the summit? As posted, I was told just the opposite by the people within the same building when it comes to starting time. I have found it best to just get up and go if you can't sleep. 2 of our last 3 we were up at 12:30 and moving towards the summit at 1:30. I, like many, a couple of things working against me sleeping the night before a summit attempt...elevation and anxiety.
Those two time were from just under 12,000' but I have problems even at 8,000' from time to time. You couple that with the excitement, sleep is very difficult to come by. So, rather than toss n' turn, go! You can always catch 40 winks once the sun comes up...and we have.
