Diamox! DMO, since you are investing so much on this trip, everyone in the group should get a prescription for Diamox from your doctor. You will likely need to educate the doctor on its use as an aid in treating/reducing altitude sickness symptoms. For AMS, people take either half or a whole 125 mg tablet twice a day, starting the day before you ascend above 10,000'. You should get the prescription, and take a day's dose while at home to see how it feels. The recommended dose is low, so you should not experience anything.
Your spending one night at 10k, and then climbing to 12k the second night is pushing your acclimatization schedule, so having the Diamox along could play a significant part in your making the summit. (After sleeping at 12k, many people wake up feeling awful.)
As for your itinerary, I'd consider staying after the hike at either the walk-in campsites at Whitney Portal, or a room at the Whitney Portal Hostel in Lone Pine (Doug at the WP Store owns it), where a bed in a shared room is about $20 a night. Then, take a leisurely drive through Death Valley, stopping at the sites you zoomed by on the way to Lone Pine. DV is a spectacular place to visit.
Edit: Search this forum for recent Diamox, threads, and Ken and Harvey are both doctors with altitude experience, so pay attention to their words:
KM &
HL
