You should sleep high, so at least Whitney Portal. I have never heard that the walk-in sites were all taken. Even if they were, it's likely that other hikers would welcome another camper or two. You definitely don't need a tent at the Portal in July.
I see from your other posts that you are spending the day before at Horseshoe Meadows. Why not camp there? Since you are overnighting at Trail Camp, you could camp at Horseshoe, drive into Lone Pine for a great breakfast, then start up the trail well fed. No need to start right at dawn.