Kennedy Meadows is nice, but last I heard you have to park your vehicle & then hike to the river & camping areas. Thats old info & maybe it has changed, been a while since I've been there. Plenty of good areas up to and in and around Bishop. But at this time of year in those areas you will almost always be around a lot of other campers.
I know you said southern Sierra's but if you have the xtra time, try north of Bishop. Great fishing & unbelievable scenery on the Walker River along 395. However it is about 3 hours north of Bishop. The good part is that the farther north you go along 395 the fewer the people & the better the views.
Have fun, I will be going myself 7/12, right after the PMC LandCruiser swap meet that same day.
John