Algorithms for joint mutual interference mitigation and object parameter estimation are a key enabler for automotive applications of frequency-modulated continuous wave (FMCW) radar. The underlying signal model poses a challenge for signal separation, since both the coherent radar echo and the non-coherent interference influenced by individual multipath propagation channels must be considered. In particular, under certain assumptions, the model is described as a superposition of multipath channels weighted by parametric chirp envelopes in the case of interference. In this paper, we introduce a method inspired by sparse Bayesian learning (SBL) to detect and estimate radar object parameters while also estimating and successively canceling the interference signal. An augmented probabilistic model is employed that uses hierarchical Gamma-Gaussian prior model for each multipath channel separately. Based on this model an iterative inference algorithm is derived using the variational expectation-maximization (EM) methodology. The algorithm is statistically evaluated in terms of object parameter estimation accuracy and robustness, indicating that it is fundamentally capable of achieving the Cramer-Rao lower bound (CRLB) with respect to the accuracy of object estimates and it closely follows the radar performance achieved when no interference is present.