Three other peaks that you might consider:
Mt Lamarck from North Lake (southwest of Bishop) is a great hike. 14 miles roundtrip, beautiful scenery, and a fun scramble at the end if you skip the Col and ascend directly up the southeast slope.
Mt Langley south of Whitney is a long hike -- 21 miles roundtrip via New Army Pass, but there are several lakes to camp by if you do an overnight. At the end, if you miss the trail to the summit, there's some fun scrambling (I'd guess easy Class 3) up the southwest side.
Another mountain in the Whitney area is Lone Pike Peak. It's about 11 miles roundtrip from the Whitney Portal, much of the way on the trail to Meysan Lake (also worth a visit, though you turn south off of the trail to ascend Lone Pike Peak about a mile before Meysan Lake). Lone Pine Peak features a chute to the summit plateau, which as 3Deserts mentioned regarding Independence, will be two steps forward, one and a half steps sliding back down on the scree. But the view from the summit, which includes Langley, Whitney, Muir, Russell, Williamson and a bunch of other mountains, is well worth it.