We present the BAT AGN Spectroscopic Survey (BASS) Near-infrared Data Release 2 (DR2), a study of 168 nearby ( z¯=0.04, z < 0.6) active galactic nuclei (AGN) from the all-sky Swift Burst Array Telescope X-ray survey observed with the Very Large Telescope (VLT)/X-shooter in the near-infrared (NIR; 0.8-2.4 μm). We find that 49/109 (45%) Seyfert 2 and 35/58 (60%) Seyfert 1 galaxies observed with VLT/X-shooter show at least one NIR high-ionization coronal line (CL; ionization potential χ > 100 eV). Comparing the emission of the [Si vi] λ1.9640 CL with the X-ray emission for the DR2 AGN, we find a significantly tighter correlation, with a lower scatter (0.37 dex) than that for the optical [O iii] λ5007 line (0.71 dex). We do not find any correlation between CL emission and the X-ray photon index Γ. We find a clear trend of line blueshifts with increasing ionization potential in several CLs, such as [Si vi] λ1.9640, [Si x] λ1.4300, [S viii] λ0.9915, and [S ix] λ1.2520, indicating the radial structure of the CL region. Finally, we find a strong underestimation bias in black hole mass measurements of Sy 1.9 using broad Hα due to the presence of significant dust obscuration. In contrast, the broad Paα and Paβ emission lines are in agreement with the M-σ relation. Based on the combined DR1 and DR2 X-shooter sample, the NIR BASS sample now comprises 266 AGN with rest-frame NIR spectroscopic observations, the largest set assembled to date.