Well John, I can't be so sure. I've never done NAP, but I've been up the Cottonwood side of AP twice, and since it is 700 feet lower I'd imagine I'd go up over that way and spend the last night in Cottonwood lakes as a last hurrah.
If I'm bumming I may sleep low at Soldier Lakes and hump up over on the last day.
I'm relatively flexible...
June might be interesting due to snow. Most folks tend to avoid Old Army Pass due to cornice that forms toward the top. I guess that Cottonwood Lakes would be more private, but there is a nice camping area on the west end of Long Lake (NAP route).
View from NAP. Long Lake is second lake

West end is on the "near" end of the lake, camping amongst the trees. The pic does not capture the scale of the camping area, or the size of the lake.

More pics available at:
http://www.whitneyzone.com/wz/ubbthreads.php/galleries/32059/Horseshoe_Meadows_-_Army_Pass_#commentsMany choices and paths to consider