Alternatively, camp near a pond and have a shot at seeing a moose.
First option would be the at Ethan Pond shelter/tent site. Never saw a moose there, but did find a future spouse, and there will surely be plenty of white-throated sparrows.
If that is crowded (expect it to be) don’t despair. Continue south a ways to the junction of Shoal Pond Trail, and turn left. Camp in the woods near (but not too near - LNT and all) once you reach the pond.
I have had real good luck with moose there and since it is a ways off the AT, you will likely have it all to yourselves.
If you elect to skip Mt Washington to simplify logistics and start in the notch, that would make for an easy first day but with an excellent landing zone — especially if you got a later start. With the Franconia Ridge ahead, you will get a true White Mountain experience. Even on of the best NH waterfalls down a very short blue blaze first thing the next day.