There is a AT shelter (Garfield Ridge Shelter. see link below) between Lafayette and Galehead just below Mt. Garfield. Do not underestimate the hike over to Garfield from Lafayette. It drops down a ways, then back up. It is possible to stealth camp south of Garfield, but youwill have to fit into small places! With that large a group you night consider stealth camping down along a side trail that leads up to Garfield from the north, getting water near the shelter first. The water for the shelter is right next to the AT just as the path to the shelter leaves the AT. Convenient if you are hiking passed.
Another option is to drop down to the 13 Falls campsite. There would plenty of room to camp there and a great water source. (the 13 falls are mini falls over a large rock!) Tent platforms, and level ground near there, too. This will add some down/up mileage.
You could find good camping spots down the Skookumchuck Trail, but you would have to lug water from Greenleaf Hut. Check for stream crossings on the Skookumchuck Trail map. Maybe there is one closer (I do not remember). But, that trail does get into woods fairly soon off the ridge. (Good way to escape bad weather on the ridge) There is a spring along the trail between the Greenleaf Hut and the summit of Lafayette.
Guyot (the Bonds) first suggestion is a good one. There are tent platforms there alongside the Guyot hut and an overflow tentsite.
That gives you an option to shorten the hike, too, by hiking out to the Lafayette Campground by a variety of trail options (Falling Waters Trail.or the Old Bridle Path via the Greenleaf Hut) by car Lafayette Campground is not too far from Lincoln Woods if you can leave a car at both places before starting your hike.
Lots of views along the loop!
Trail( AT) from Garfield Hut to So. Twin is one steep mile! No turns.