Estimating the rigid transformation between two LiDAR scans through putative 3D correspondences is a typical point cloud registration paradigm. Current 3D feature matching approaches commonly lead to numerous outlier correspondences, making outlier-robust registration techniques indispensable. Many recent studies have adopted the branch and bound (BnB) optimization framework to solve the correspondence-based point cloud registration problem globally and deterministically. Nonetheless, BnB-based methods are time-consuming to search the entire 6-dimensional parameter space, since their computational complexity is exponential to the solution domain dimension in the worst-case. To enhance algorithm efficiency, existing works attempt to decouple the 6 degrees of freedom (DOF) original problem into two 3-DOF sub-problems, thereby reducing the search space. In contrast, our approach introduces a novel pose decoupling strategy based on residual projections, decomposing the raw registration problem into three sub-problems. Subsequently, we embed interval stabbing into BnB to solve these sub-problems within a lower two-dimensional domain, resulting in efficient and deterministic registration. Moreover, our method can be adapted to address the challenging problem of simultaneous pose and registration. Through comprehensive experiments conducted on challenging synthetic and real-world datasets, we demonstrate that the proposed method outperforms state-of-the-art methods in terms of efficiency while maintaining comparable robustness.