I remember Bob R sometimes posted that there are some quiet and more secluded spots north of the Trail Camp pond. On the satellite view, I can see a number of camping pads just north of the outlet of the pond. I think you would need to climb just a little to get to them.
Regarding your group's desire for acclimatization: Climbing all the way to 12.5k (Trail Camp) and spending the night (only 12 hours) might do the opposite: bring on enough AMS symptoms that you feel rotten in the morning. Staying at Outpost is a more moderate elevation, so might not be so severe a shock to the system. If I were in your group, I'd be lobbying to spend the night before you start hiking at 10k at Horseshoe Meadows.
