Using a suite of numerical calculations, we consider the long-term evolution of
circumbinary debris from the Pluto--Charon giant impact. Initially, these solids
have large eccentricity and pericenters near Charon's orbit. On time scales of
100--1000 yr, dynamical interactions with Pluto and Charon lead to the ejection
of most solids from the system. As the dynamics moves particles away from the
barycenter, collisional damping reduces the orbital eccentricity of many particles.
These solids populate a circumbinary disk in the Pluto-Charon orbital plane; a large
fraction of this material lies within a `satellite zone' that encompasses the orbits
of Styx, Nix, Kerberos, and Hydra. Compared to the narrow rings generated from the
debris of a collision between a trans-Neptunian object (TNO) and Charon,
disks produced after the giant impact are much more extended and may be a less promising option for producing small circumbinary satellites.
We consider a scenario where the small satellites of Pluto and Charon grew within a disk of debris from an impact between Charon and a trans-Neptunian object (TNO). After Charon's orbital motion boosts the debris into a disk-like structure, rapid orbital damping of meter-sized or smaller objects is essential to prevent the subsequent reaccretion or dynamical ejection by the binary. From analytical estimates and simulations of disk evolution, we estimate an impactor radius of 30-100 km; smaller (larger) radii apply to an oblique (direct) impact. Although collisions between large TNOs and Charon are unlikely today, they were relatively common within the first 0.1-1 Gyr of the solar system. Compared to models where the small satellites agglomerate in the debris left over by the giant impact that produced the Pluto-Charon binary planet, satellite formation from a later impact on Charon avoids the destabilizing resonances that sweep past the satellites during the early orbital expansion of the binary.