We present two independent measurements of stellar velocity dispersions (σ⋆) from the Ca ii H+K λ3969, 3934 and Mg ibλ 5183, 5172, 5167 region (3880–5550 Å) and the calcium triplet region (8350–8750 Å) for 173 hard X-ray-selected Type 1 active galactic nuclei (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 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 supermassive black hole 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. We also find an anticorrelation 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.