Nothing wrong with LP Campground - but you won't really get any acclimatization sleeping there. Like wbtravis stated, there's usually plenty of non-reservable sites available at the Portal, and the one-night max backpacker sites are completely non-reservable, so if you're there fairly early those are usually up for grabs as well. Last year we reserved 1 night at the Portal and moved the second night to the backpacker sites. It made for an easy pack-up before the hike.
Another option (although about a 45 min to 1 hr drive away) is the one night campgrounds at the Cottonwood Lakes & Horeshoe Meadow trailheads (also non-reservable, but I've never seen them full). A long, curvy drive away, but you can get a night of acclimatization at 10,000 feet.