This paper presents a Bayesian reformulation of covariate-assisted principal regression for covariance matrix outcomes to identify low-dimensional components in the covariance associated with covariates. By introducing a geometric approach to the covariance matrices and leveraging Euclidean geometry, we estimate dimension reduction parameters and model covariance heterogeneity based on covariates. This method enables joint estimation and uncertainty quantification of relevant model parameters associated with heteroscedasticity. We demonstrate our approach through simulation studies and apply it to analyze associations between covariates and brain functional connectivity using data from the Human Connectome Project.