I am pretty sure I have seen places to pitch a tent along the 19 Mile Brook Trail just outside of the FPA, or maybe on the Wildcat Ridge section of the AT in that same general area.
Not 100% sure though - never camped there myself.
One thing to keep in mind is that so long as you are 1/4 mile from the hut, there is no requirement that you head off the tail 200 feet in that area (it is not a designated wilderness) — so you could camp at a spot right along the AT or the 19 Mile Brook Trail if you found one.
Finding sites (albeit marginal ones) just outside a 1/4 mile FPA is rather common, as it is at Trail junctions.
Another thing to keep in mind is that camping at spots like that will stick in the craw of some folks — even when they are perfectly legal places, they may not realize iit.