Coupled multiscale simulations require thousands of simulations on representative volume elements (RVEs) subject to different boundary conditions. In this parametric multi-query setting, projection-based model reduction (MOR) methods and hyperreduction techniques can be deployed to reduce runtimes by multiple orders of magnitude [1]. In particular, researchers have recently explored nonlinear approximation spaces with the aim to represent the solution manifold more tightly than the approximation spaces obtained via the proper orthogonal decomposition can. The local basis method does so via multiple, localised POD models [2], while polynomial manifold approaches exploit a continuously nonlinear polynomial Ansatz [3]. Neural networks can be used to construct approximation spaces more flexibly, but require considerable amounts of training data [4]. In a recent work, we employed a graph-based manifold learning approach which constructs flexible embeddings, but is comparatively data-prudent [1]. In this contribution, we exploit several methods for the construction of nonlinear approximation spaces and compare their performance on a quasi-static solid-mechanical RVE problem. We deploy several popular hyperreduction techniques in tandem with these, and explore the accuracy and runtime which can be achieved over a range of parameters. We observe that nonlinear MOR techniques can yield overall performance gains in the tradeoff between accuracy and online runtime. We emphasise that this is only the case if the smaller approximation spaces obtained via nonlinear MOR techniques also facilitate the construction of smaller hyperreduction models. [1]L. Scheunemann and E. Faust. “A manifold learning approach to nonlinear model order reduction of quasi-static problems in solid mechanics: proof of concept and parameter study”. In: Computational Mechanics (Dec. 2025). [2]D. Amsallem, M. J. Zahr, and C. Farhat. “Nonlinear Model Order Reduction Based on Local Reduced-order Bases”. In: International Journal for Numerical Methods in Engineering 92.10 (Dec. 2012). [3]J. Barnett and C. Farhat. “Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction”. In: Journal of Computational Physics 464 (Sept. 2022). [4]J. Barnett, C. Farhat, and Y. Maday. “Neural-Network-Augmented Projection-Based Model Order Reduction for Mitigating the Kolmogorov Barrier to Reducibility”. In: Journal of Computational Physics 492 (Nov. 2023)