Virial black hole (BH) mass (MBH) determination directly involves knowing the broad-line region (BLR) clouds' velocity distribution, their distance from the central supermassive BH (RBLR), and the virial factor (f). Understanding whether biases arise in MBH 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{mu}m) 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 H{alpha} and near-infrared broad lines (HeI, Pa{beta}, Pa{alpha}) remain unbiased to either BLR extinction or X-ray obscuration, the H{alpha} broad-line luminosity is suppressed when N_H_>~10^21^cm^-2^, systematically underestimating MBH by 0.23-0.46dex. Near-infrared line luminosities should be preferred to H{alpha} until N_H_<10^22^cm^-2^, while at higher obscuration a less-biased RBLR proxy should be adopted. We estimate f for Seyfert 1 and 2 using two obscuration-unbiased MBH 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 MBH. 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.