Algorithms for mutual interference mitigation and object parameter estimation are a key enabler for automotive applications of frequency-modulated continuous wave (FMCW) radar. In this paper, we introduce a signal separation method to detect and estimate radar object parameters while jointly estimating and successively canceling the interference signal. The underlying signal model poses a challenge, since both the coherent radar echo and the non-coherent interference influenced by individual multipath propagation channels must be considered. Under certain assumptions, the model is described as a superposition of multipath channels weighted by parametric interference chirp envelopes. Inspired by sparse Bayesian learning (SBL), we employ an augmented probabilistic model that uses a hierarchical Gamma-Gaussian prior model for each multipath channel. Based on this, 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.