"From Piute, you will have to make it all the way to the JMT to hit 8200' Near the trail junction there is a cluster of heavily used sites. There are also good sites in French Canyon and acres of open space at 11k."
Paiute is probably the easiest pass on the east side, at least from the Bishop area on south, and it is all easy downhill from the pass to JMT. There are loads of good campsites strung out along Paiute Creek, and if you waited until you were just above the final descent to JMT, you could camp at or just below 10,000', IIRC. If you are feeling good, a delightful place to camp would be to cross Paiute Creek and hike the use trail up to Honeymoon Lake. It is an exquisite little lake in a gorgeous setting, and is not often used.