There is boondocking in the volcanic tablelands north of Bishop. (This area can get hot, though.) When you say "holiday," is that Labor Day?
I also like the Bristlecone area, but let me give you an alternative idea -- the Bishop Creek area. There are lots and lots of campgrounds up there -- it is almost never completely full. Plus, there is a lot of hiking up there, while Bristlecone has fewer trails. There is some boondocking not too far away in the Buttermilk Country area, but the campgrounds are at higher elevations and are cooler.
The other advantage of the Bishop Creek campgrounds is that many of them are right on the creek -- lots of things for three active kids to do!
