We examine the internal properties of the most massive ultracompact dwarf galaxy (UCD), M59-UCD3, by combining adaptive-optics-assisted near-IR integral field spectroscopy from Gemini/NIFS and Hubble Space Telescope (HST) imaging. We use the multiband HST imaging to create a mass model that suggests and accounts for the presence of multiple stellar populations and structural components. We combine these mass models with kinematics measurements from Gemini/NIFS to find a best-fit stellar mass-to-light ratio (M/L) and black hole (BH) mass using Jeans anisotropic models (JAMs), axisymmetric Schwarzschild models, and triaxial Schwarzschild models. The best-fit parameters in the JAM and axisymmetric Schwarzschild models have BHs between 2.5 and 5.9 million solar masses. The triaxial Schwarzschild models point toward a similar BH mass but show a minimum {chi}^2^ at a BH mass of ~0. Models with a BH in all three techniques provide better fits to the central V_rms_ profiles, and thus we estimate the BH mass to be 4.2_-1.7_^+2.1^x10^6^M_{sun}_ (estimated 1{sigma} uncertainties). We also present deep radio imaging of M59-UCD3 and two other UCDs in Virgo with dynamical BH mass measurements, and we compare these to X-ray measurements to check for consistency with the fundamental plane of BH accretion. We detect faint radio emission in M59cO but find only upper limits for M60-UCD1 and M59-UCD3 despite X-ray detections in both these sources. The BH mass and nuclear light profile of M59-UCD3 suggest that it is the tidally stripped remnant of a ~10^9^-10^10^M_{sun}_ galaxy.