We present two independent measurements of stellar velocity dispersions ( σ⋆ ) from the Ca\,H+K \& Mg\,\textsc{i} region (3880--5550~\AA) and the Calcium Triplet region (CaT, 8350--8750~\AA) for 173 hard X-ray-selected Type 1 AGNs (z≤ 0.08) from the 105-month Swift-BAT catalog. We construct one of the largest samples of local Type 1 AGNs that have both single-epoch (SE) 'virial' black hole mass (MBH) estimates and σ⋆ measurements obtained from high spectral resolution data, allowing us to test the usage of such methods for SMBH studies. We find that the two independent σ⋆ measurements are highly consistent with each other, with an average offset of only 0.002±0.001 dex. Comparing MBH estimates based on broad emission lines and stellar velocity dispersion measurements, we find that the former is systematically lower by ≈0.12 dex. Consequently, Eddington ratios estimated through broad-line MBH determinations are similarly biased (but in the opposite way). We argue that the discrepancy is driven by extinction in the broad-line region (BLR). We also find an anti-correlation between the offset from the MBH - σ⋆ relation and the Eddington ratio. Our sample of Type 1 AGNs shows a shallower MBH - σ⋆ relation (with a power law exponent of ≈3.5) compared with that of inactive galaxies (with a power-law exponent of ≈4.5), confirming earlier results obtained from smaller samples.