I second Laura's advice on camping at Grandview. That will give you some acclimation before you drive on up to the Barcroft gate. Granted, it's an hour drive on to the gate, but I wouldn't drive strait to the 12,000 ft elev and camp -- someone in your party might have trouble the next day with the hike. ...but then, if you do one of those SoCal mountains the weekend before, that will get your acclimation started. So it might just work. You really don't know for sure until you've been to 14K, though.
Grandview is a first-come-first-serve campground, and can fill up in the summer, so that might be an issue. If full, you could just go for it and camp at the Barcroft gate, with the awareness that hiking to White Mtn might be give you some AMS symptoms.
Take lots of water -- there is none in the White mountains at all.
If you can spend a second night at Grandview, you'd be better acclimated for Whitney. But then if nobody has problems climbing White, then you're good -- head on down to Bishop.
I think the day between the White Mountain climb and the Whitney climb is the perfect amount of time to rest. Once you get your permits, you could even drive up to Horseshoe Meadows just to see the drive -- about 45 minutes each way from Lone Pine. The hike on the Main Trail up to Lone Pine Lake is a nice idea, but ONLY if you are not too wasted from the White Mtn hike.
There are a number of places to stay in Lone Pine. The Whitney Portal Hostel is least expensive ($20 per bed if you share a room with others). Also check the
lodging link within the Orientation Notes.
Be sure to have a good time when you go. And post a trip report when you're done. We always like to read what worked for you.
