Virial black hole (BH) mass (M BH) determination directly involves knowing the broad-line region (BLR) clouds' velocity distribution, their distance from the central supermassive BH (R BLR), and the virial factor (f). Understanding whether biases arise in M BH estimation with increasing obscuration is possible only by studying a large (N > 100) statistical sample of obscuration-unbiased (hard) X-ray-selected active galactic nuclei (AGNs) in the rest-frame near-infrared (0.8-2.5 μm) since it penetrates deeper into the BLR than the optical. We present a detailed analysis of 65 local Burst Alert Telescope (BAT) selected Seyfert galaxies observed with Magellan/FIRE. Adding these to the near-infrared BAT AGN spectroscopic survey database, we study a total of 314 unique near-infrared spectra. While the FWHMs of Hα and near-infrared broad lines (He I, Paβ, Paα) remain unbiased to either BLR extinction or X-ray obscuration, the Hα broad-line luminosity is suppressed when N H ≳ 1021 cm-2, systematically underestimating M BH by 0.23-0.46 dex. Near-infrared line luminosities should be preferred to Hα until N H < 1022 cm-2, while at higher obscuration a less-biased R BLR proxy should be adopted. We estimate f for Seyfert 1 and 2 using two obscuration-unbiased M BH measurements, i.e., the stellar velocity dispersion and a BH mass prescription based on near-infrared and X-ray, and find that the virial factors do not depend on the redshift or obscuration, but some broad lines show a mild anticorrelation with M BH. Our results show the critical impact obscuration can have on BLR characterization and the importance of the near-infrared and X-rays for a less-biased view of the BLR. *This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile.