Multi-group Bayesian structural equation modeling (MG-BSEM) gained considerable attention among substantive researchers investigating cross-group differences and methodologists exploring challenges in measurement invariance testing. MG-BSEM allows for greater flexibility by applying elastic rather than strict equality constraints on item parameters across groups. This, however, requires a specification of user-defined prior variances for cross-group differences in item parameters. Although prior selection in general Bayesian settings is well-studied, guidelines with respect to tuning the normal prior variances in MG-BSEM approximate measurement invariance (AMI) analysis are still largely missing. In a Monte Carlo simulation study we find that correctly specifying prior variances results in more precise credibility intervals (CI) and posterior standard deviations, while prior misspecification has little influence on point estimates. We compared the BIC, DIC, and PPP fit measures and found in our simulation scenarios that the DIC measure was most effective, when a proper threshold for model selection was applied.