There are no field brown dwarf analogs with measured masses, radii, and luminosities, precluding our ability to connect the population of transiting brown dwarfs with measurable masses and radii and field brown dwarfs with measurable luminosities and atmospheric properties. LHS 6343 C, a weakly irradiated brown dwarf transiting one member of an M+M binary in the Kepler field, provides the first opportunity to probe the atmosphere of a non-inflated brown dwarf with a measured mass and radius. Here, we analyze four Spitzer observations of secondary eclipses of LHS 6343 C behind LHS 6343 A. Jointly fitting the eclipses with a Gaussian process noise model of the instrumental systematics, we measure eclipse depths of 1.06 ±0.21 ppt at 3.6 μm and 2.09 ±0.08 ppt at 4.5 μm, corresponding to brightness temperatures of 1026 ±57 K and 1249 ±36 K, respectively. We then apply brown dwarf evolutionary models to infer a bolometric luminosity . Given the known physical properties of the brown dwarf and the two M dwarfs in the LHS 6343 system, these depths are consistent with models of a 1100 K T dwarf at an age of 5 Gyr and empirical observations of field T5-6 dwarfs with temperatures of 1070 ±130 K. We investigate the possibility that the orbit of LHS 6343 C has been altered by the Kozai-Lidov mechanism and propose additional astrometric or Rossiter-McLaughlin measurements of the system to probe the dynamical history of the system.