Real-time simulation of engineering problems that contain parametrized geometries is critical for interactive design and digital twin applications. While model order reduction (MOR) [1] effectively accelerates solvers for material property parameterization, shape parametrization poses unique challenges, particularly when geometric constraints must be strictly preserved during deformation; e.g., regions containing pins must retain their shape during deformation. Standard mesh warping or topological parametrization techniques often fail to maintain these constrained subdomains or lack the computable, invertible Jacobians required for efficient MOR in a reference-domain formulation. This work presents a novel geometry transformation framework that enables MOR for shape-parametrized geometries while strictly enforcing rigid body motion for prescribed constrained subdomains. The approach utilizes a hierarchical free-form deformation (FFD) strategy: a coarse grid parametrizes global low-dimensional shape changes (e.g., bending, twisting), while a fine grid enables localized control. We introduce a least-squares rigidization procedure [2] that locally overrides the FFD interpolation in constrained regions, computing the optimal rigid transformation to preserve the exact geometry of interfaces while ensuring C1 continuity across the domain. Crucially, this composite transformation is integrated into a reference-domain finite element formulation, which enables the construction of a geometrically consistent reduced basis space, and enables hyperreduction by the empirical quadrature procedure [3] to achieve online-efficiency. The framework is demonstrated on industrial case studies, including an excavator arm and an automotive suspension rocker. Results indicate the method captures complex non-affine deformations and preserves rigid regions with less than 1% error in simulated quantities of interest, offering a robust pathway for real-time structural analysis of complex assemblies. REFERENCES [1] A. T. Patera and G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2006. [2] S. Umeyama, "Least-squares estimation of transformation parameters between two point patterns," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 4, pp. 376–380, 1991. [3] M. Yano and A.T. Patera. An LP empirical quadrature procedure for