We present the first direct measurement of the central black hole mass, M •, in NGC 6086, the Brightest Cluster Galaxy (BCG) in A2162. Our investigation demonstrates for the first time that stellar-dynamical measurements of M • in BCGs are possible beyond the nearest few galaxy clusters. We observed NGC 6086 with laser guide star adaptive optics and the integral-field spectrograph (IFS) OSIRIS at the W. M. Keck Observatory and with the seeing-limited IFS GMOS-N at Gemini Observatory North. We combined the IFS data sets with existing major-axis kinematics and used axisymmetric stellar orbit models to determine M • and the R-band stellar mass-to-light ratio, M /LR . We find M • = 3.6+1.7 –1.1 × 109 M ☉ and M /L R = 4.6+0.3 –0.7 M ☉ L ☉ –1 (68% confidence) from models using the most massive dark matter halo allowed within the gravitational potential of the host cluster. Models fitting only IFS data confirm M • ~ 3 × 109 M ☉ and M /LR ~ 4 M ☉ L ☉ –1, with weak dependence on the assumed dark matter halo structure. When data out to 19 kpc are included, the unrealistic omission of dark matter causes the best-fit black hole mass to decrease dramatically, to 0.6 × 109 M ☉, and the best-fit stellar mass-to-light ratio to increase to 6.7 M ☉ L –1 ☉,R . The latter value is at further odds with stellar population studies favoring M /LR ~ 2 M ☉ L –1 ☉. Biases from dark matter omission could extend to dynamical models of other galaxies with stellar cores, and revised measurements of M • could steepen the empirical scaling relationships between black holes and their host galaxies.