If you could camp higher, at least 8k, you would be getting some better acclimatization in. I would try for 2 nights at 8k or above before hiking the Whitney trail. But you should drive through Death Valley if you've not seen it.
It is extra driving north, but the Hwy 168 over the White Mountain area (Westgard Pass) puts you close to the Bristlecone Pine area, which is a worthwhile sight seeing side trip. Grand View campground is dry but at 8k.
Horseshoe Meadows off the Whitney Portal road gets you to 10k and plenty of camping, too.
