Congratulations on getting the permit online! Your persistence paid off.
In the past, Inyo N.F. would put permits in the night-drop box in the kiosk out by the road in front of the Visitor Center, but I don't know if that is still allowed. Call the Wilderness Permit office and ask: 760-873-2483 (and report back what you find out, please!)
If they won't let you use the night-drop, here is what I would do: Drive up to Horseshoe Meadows and spend the night. Doing that puts you at 10k elevation, to get your acclimation going. Just starting up the trail, with an overnight pack, without any acclimation can bring on AMS 24 hours later. In fact, consider camping at Outpost camp and starting your summit day early, rather than camping at Trail Camp.
Even if you DO get the night-drop permit, still consider the higher HM acclimation overnight. It only takes 40 minutes to drive back down to Lone Pine.
As for the solo hike: You will NOT be alone. Especially where you spend the night. There will be over 200 people using the trail that weekend. If you're friendly, you will likely find people hiking about your same pace to hike with.
Here's a good thread:
Hiking Whitney alone
