Grandview campground is just 5 miles from the Schulman Grove visitor center, on the road up to White Mountain Station. It would give you a night or two to acclimate without having to descend back down to the Lone Pine altitude. Campground altitude is 8,600 feet.
https://www.fs.usda.gov/recarea/inyo/recarea/?recid=20268Aside from having to bring all your own water, it is clean and has toilets. No reservations, but plenty of first come first served campsites. I have stayed there many times and found it to be a good base for White Mountain hikes. As it's name implies, it provides wonderful panoramic views of the Sierras. And no bears!
