Keep in mind that there are campsites at both the Cottonwood Lakes Trailhead and at Cottonwood Pass Trailhead.
As for the sites filling up fast, being Fourth of July weekend, there will be lots of people there.
Consider staying at Whitney Portal Ravine campsites. You access the sites via the overflow parking lot. Much like the walkin sites at Horseshoe Meadows, you carry your equipment down to the campsites. However, keep in mind that access to water is at the overflow parking lot as well as the restrooms.
The sites have raised tent platforms, bear boxes and fire pits. As for the one-night stay, I have camped ther multiple nights, and have seen many campers there over several days. There are rarely more than a few people there, I believe there are 8 campsites.
https://www.fs.usda.gov/recarea/inyo/recarea/?recid=20704paul