We explore the possibility that GSC 07396-00759 (spectral type M1e) is a widely separated (~2.82', or projected separation ~12,350 AU) companion to the "old" (age ~12 Myr) classical T Tauri binary system V4046 Sgr AB, as suggested by the proximity and similar space motions of the two systems. If the two systems are equidistant and coeval, then GSC 07396--00759, like V4046 Sgr AB, must be a spectroscopic binary with nearly equal-mass components, and V4046 Sgr must be at least ~8 Myr old. Analysis of a serendipitous Chandra X-ray gratings spectrum and light curve as well as XMM-Newton light curves and CCD spectra of GSC 07396-00759 obtained during long exposures targeting V4046 Sgr AB reveals a relatively hard (T_X ~ 10^7 K) X-ray spectrum, strong flaring, and relatively low-density plasma. These X-ray characteristics of GCS 07396--00759 are indicative of a high level of coronal activity, consistent with its apparent weak-lined T Tauri star status. Interactions between V4046 Sgr AB and GCS 07396-00759 when the two systems were more closely bound may be responsible for (a) their dissolution ~10^6 yr ago, (b) the present tight, circular orbit of V4046 Sgr AB, and (c) the persistence of the gaseous circumbinary disk still orbiting V4046 Sgr AB. ; Comment: 14 pages, 3 figures; to appear in the Astrophysical Journal (Letters)