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We present Bayesian AGN Decomposition Analysis for SDSS Spectra (BADASS), an open source spectral analysis code designed for automatic detailed deconvolution of AGN and host galaxy spectra, implemented in Python, and designed for the next generation of large scale surveys. BADASS simultaneously fits all spectral components, including power-law continuum, stellar line-of-sight velocity distribution, FeII emission, as well as forbidden (narrow), permitted (broad), and outflow emission line features, all performed using Markov Chain Monte Carlo to obtain robust uncertainties and autocorrelation analysis to assess parameter convergence. BADASS utilizes multiprocessing for batch fitting large samples of spectra while efficiently managing memory and computation resources and is currently being used in a cluster environment to fit thousands of SDSS spectra. We use BADASS to perform a correlation analysis of 63 SDSS type 1 AGNs with evidence of strong non-gravitational outflow kinematics in the [OIII]$\lambda5007$ emission feature. We confirm findings from previous studies that show the core of the [OIII] profile is a suitable surrogate for stellar velocity dispersion $σ_*$, however there is evidence that the core experiences broadening that scales with outflow velocity. We find sufficient evidence that $σ_*$, [OIII] core dispersion, and the non-gravitational outflow dispersion of the [OIII] profile form a plane whose fit results in a scatter of $\sim0.1$ dex. Finally, we discuss the implications, caveats, and recommendations when using the [OIII] dispersion as a surrogate for $σ_*$ for the $M_{\rm{BH}}-σ_*$ relation.