Multipath-based simultaneous localization and mapping (MP-SLAM) is a promising approach for future 6G networks to jointly estimate the positions of transmitters and receivers together with the propagation environment. In cooperative MP-SLAM, information collected by multiple mobile-terminals (MTs) is fused to enhance accuracy and robustness. Existing methods, however, typically assume perfectly synchronized base stations (BSs) and orthogonal transmission sequences, rendering inter-BS interference at the MT negligible. In this work, we relax these assumptions and address simultaneous source separation, synchronization, and mapping. A relevant example arises in modern 5G systems, where BSs employ muting patterns to mitigate interference, yet localization performance still degrades. We propose a novel BS-dependent data association and synchronization bias model, integrated into a joint Bayesian framework and inferred via the sum-product algorithm on a factor graph. The impact of joint synchronization and source separation is analyzed under various system configurations. Compared with state-of-the-art cooperative MP-SLAM assuming orthogonal and synchronized BSs, our statistical analysis shows no significant performance degradation. The proposed BS-dependent data association model constitutes a principled approach for classifying features by arbitrary properties that persist over time, such as reflection order or feature type (scatter points versus walls).