Well, the "fellas" might give you different advice, but if you'd like one woman's POV.
I don't think there is actually camping allowed at Schulman Grove, nor at the other groves in the forest. Please see the Inyo NF
page page for more information, but it looks like Grandview CG (8560ft) is as high as you're going to get. Frankly, coming from sea level it's a good option. Make sure to bring plenty of water, as the only facilities available are vault toilets.
Take your time wandering around the Pine groves. You, or others in your group, may decide to do this instead of walking the long road to White. They are spectacular trees, and, quite frankly, offer wonderful places for cozy naps high above the rest of the world.
If you do decide to walk up White, take your time and enjoy the views. The road is spectacular as it winds along the ridge. There are a lot of ups and downs, which give you a chance to feel your body work differently as you gain altitude. Because it isn't a particularly difficult walk (aside from the altitude factor), you might want to take a little backpacker's stove and make some hot soup or a treat on the summit.
No matter what, keep it relatively easy and get used to being high. Taking the extra day at the portal would be great, too, if it fits with your schedule. You might even consider a stroll up the Meysan Lake trail to see another drainage while you're there acclimating.
Have a great trip!
-L
