Young giant exoplanets are a unique laboratory for understanding cool, low-gravity atmospheres. A quintessential example is the massive extrasolar planet β Pic b, which is 9 AU from and embedded in the debris disk of the young nearby A6V star β Pictoris. We observed the system with first light of the Magellan Adaptive Optics (MagAO) system. In Paper I (Males et al. 2014) we presented the first CCD detection of this planet with MagAO+VisAO. Here we present four MagAO+Clio images of β Pic b at 3.1 µm, 3.3 µm, L ′ , and M ′ , including the first observation in the fundamental CH 4 band. To remove systematic errors from the spectral energy distribution (SED), we re-calibrate the literature photometry and combine it with our own data, for a total of 22 independent measurements at 16 passbands from 0.99-4.8 µm. Atmosphere models demonstrate the planet is cloudy but are degenerate in effective temperature and radius. The measured SED now covers >80% of the planet's energy, so we approach the bolometric luminosity empirically. We calculate the luminosity by extending the measured SED with a blackbody and integrating to find log(L bol /L ⊙ ) = −3.78 ± 0.03. From our bolometric luminosity and an age of 23±3 Myr, hot-start evolutionary tracks give a mass of 12.7±0.3 M Jup , radius of 1.45±0.02 R Jup , and T eff of 1708±23 K (model-dependent errors not included). Our empirically-determined luminosity is in agreement with values from atmospheric models (typically −3.8 dex), but brighter than values from the field-dwarf bolometric correction (typically −3.9 dex), illustrating the limitations in comparing young exoplanets to old brown dwarfs.