I took a chance and drove to the Lone Pine Campground just below Mt. Whitney on Memorial Day weekend of 2017. After circling the campground twice I asked the campground manager if there was an availability. After double checking his paperwork I managed to snag a site 43 which was at the far end at the top of the grounds.

This site is meant for group camping and it really is quite large with a picnic table made for 20 people to sit at. There is a fire ring to accommodate a very large fire. It may have been 4 feet across, I didn't have a tape measure.

The drawback of this site is that you will have to walk everything at least 100 feet uphill. Not ideal for a lot of comfort gear. On the plus side, the bathroom isn't far away and they are clean.