We present the BAT AGN Spectroscopic Survey (BASS) Near-infrared Data Release 2 (DR2), a study of 168 nearby (\bar{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{mu}m). 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 {chi}>100eV). Comparing the emission of the [SiVI]{lambda}1.9640 CL with the X-ray emission for the DR2 AGN, we find a significantly tighter correlation, with a lower scatter (0.37dex) than that for the optical [OIII]{lambda}5007 line (0.71dex). We do not find any correlation between CL emission and the X-ray photon index {Gamma}. We find a clear trend of line blueshifts with increasing ionization potential in several CLs, such as [SiVI]{lambda}1.9640, [SiX]{lambda}1.4300, [SVIII]{lambda}0.9915, and [SIX]{lambda}1.2520, 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 H{alpha} due to the presence of significant dust obscuration. In contrast, the broad Pa{alpha} and Pa{beta} emission lines are in agreement with the M-{sigma} 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.