In the nonlinear regime, the efficiency of multiscale methods depends critically on reduced-order modeling and, in particular, on hyper-reduction techniques for the evaluation of internal forces. This requirement arises, for example, in the Empirical Interscale Finite Element Method (EIFEM) [1], where heterogeneous structures are decomposed into subdomains treated as superelements and fine-scale nonlinear responses are mapped directly onto coarse-scale variables through a variational interscale formulation. In this setting, the internal response of each subdomain can be efficiently approximated using nonlinear-manifold hyper-reduced order models (HROMs), in which the solution is expressed in terms of a low-dimensional set of latent coordinates. The effectiveness of nonlinear-manifold HROMs depends crucially on the construction of accurate and robust hyper-reduction schemes. Standard empirical cubature methods employ integration weights that are fixed over the entire solution manifold, despite the fact that the integrand evolves smoothly on a low-dimensional nonlinear manifold. This limitation may lead to suboptimal sparsification and robustness issues in nonlinear solution procedures. To overcome this limitation, we introduce the Manifold Adaptive Weight Empirical Cubature Method (MAW-ECM), which builds upon the approach presented in [3] by allowing cubature weights to adapt smoothly along the nonlinear solution manifold. MAW-ECM constructs sparse cubature rules through a greedy elimination of integration points combined with a constrained weight redistribution, while smoothness across the manifold is enforced via graph-based regularization. Numerical results in multiscale analysis show that the number of selected integration points scales with the intrinsic manifold dimension, yielding reductions of up to three orders of magnitude in computational cost compared to standard HROMs, while preserving accuracy and robustness.