With that much time to acclimate, at least some should be spent higher, at Horseshoe Meadows. One night at a time there, and it will do you more good than at WP.
There are a number of first come camping spots. Sleeping in vehicles is frowned upon but not often busted, but may not be possible at all with the construction going on. Don't count on it. Grab a campsite, and plan to pick up others in Lone Pine. There is likely to be no parking at the portal for overnighters.
0500 leaves plenty of time to get to Trail Camp: its 6 miles in, and a mile an hour is a pretty common pace on the trail.
