Identifying AGNs from X-ray detections$-$II: Metallicity calibrations for the $\rm N_2O_2$ and $\rm N_2S_2$ diagnostics
Emission-line ratios from ions with nearly identical ionization potentials offer a robust solution to the degeneracies inherent to traditional active galactic nuclei (AGNs) metallicity diagnostics. We introduce new semi-empirical metallicity calibrations for the $\rm N_2O_2$ and $\rm N_2S_2$ diagnostics, explicitly designed to isolate the chemical abundance from the incident radiation field. By coupling an extensive grid of CLOUDY photoionization simulations directly to the intrinsic 2$-$10 keV X-ray luminosity ($L_{\rm X}$) and benchmarking against Seyfert~2 nuclei from the Burst Alert Telescope AGN Spectroscopic Survey (BASS), we establish robust relations valid across the metallicity regime of $8.0 \lesssim 12+\log({\rm O/H}) \lesssim 9.1$ ($0.2 \lesssim Z/Z_{\odot} \lesssim 2.6$). The $\rm N_2O_2$ and $\rm N_2S_2$ indices trace co-spatial emitting volumes within the narrow-line regions (NLRs), enabling this framework to resolve the significant $L_{\rm X}$-driven systematic biases we previously identified in the standard $\rm N_2$ and $\rm N_2O_3$ indices. While the $\rm N_2O_2$ ratio proves to be virtually independent of nebular structural variations, the $\rm N_2S_2$ index exhibits a subtle, yet discernible, electron density ($N_{\rm e}$) susceptibility due to the low critical density ($N_{\rm c}$) of the [S II]$\lambda\lambda6716,6731$ doublet. Nevertheless, both diagnostics yield highly precise metallicity constraints with tight root-mean-square residual dispersions of $\sim 0.081$ dex for $\rm N_2O_2$ and $\sim 0.121$ dex for $\rm N_2S_2$. We propose the $\rm N_2O_2$ and $\rm N_2S_2$ calibrations as highly optimized, unbiased metallicity tracers for AGNs.