Cirque Peak (12,900') would be good.
I've gone over Cottonwood Pass in June in big snow years 3 times. Twice without any gear. The pass has a less exposed back door, just look at the Cirque Peak 7.5' Quad. It is very easy to figure out.
Continue north on the PCT through Chicken Spring Lake. After the climb out, when the uphill side of the trail opens up, angle to the peak. There will be some snow field, which can be avoided or crossed, I chose the latter. At the end, there is some bouldering to the summit...only exposed if you chose a route with exposure.
This is about 16-miles RT from the Cottonwood Pass Trailhead.
