The next exit point south is Horseshoe Meadows. I think that adds 2 or 3 days hiking from Crabtree, via the PCT, past Chicken Spring Lake, over Cottonwood Pass. Though I haven't been that way yet, it seems a bit less spectacular after all the fun stuff coming in from Kearsarge. It might be more enjoyable to go south to north if you include Cottonwood.
You would have the option of also exiting over Army Pass or New Army Pass -- slightly higher than Cottonwood Pass, but you get to see Cottonwood Lakes.
Exiting Cottonwood is easier than Trail Crest, though, since you won't need to haul your packs up and over such a high pass.
I have the feeling you will be able to get your permits to exit Whitney. If not, then get the Kearsarge entry without the exit permit, then (Plan B) try for some walk-in exit permits when you pick up the permit. Plan C would then be to extend the trip and exit at Horseshoe.
Plan D could be exiting via the MR and the "Easy Walk-off" route, but I don't think everyone in your group would enjoy that route, and it requires carrying the backpacks all the way to the Whitney summit.
BTW, before you actually get to the time when you want to reserve your actual permit, you NEED to get online at recreation.gov and practice entering a trip. It gives you a 15-minute window to setup the trip, and that includes specifying overnight locations each night of the trip. You can put in bogus camp locations and change them later, but you cannot change the entry date and exit permit date at all, so get the important part done and pay first. On your actual trip, the overnight locations are not something you MUST adhere to. They are only for statistical info and starting points if a SAR is necessary.
Practice link
