Nonlinear finite element simulations are central to modern engineering, but they remain expensive to run. This has motivated the development of hyper-reduction methods such as the Discrete Empirical Interpolation Method (DEIM) and Energy Conserving Sampling and Weighting (ECSW). Each comes with its own limitations. In highly nonlinear problems, especially those involving exponential terms, DEIM often requires a large number of modes and evaluation points to achieve acceptable accuracy. ECSW, on the other hand, demands significant memory during the offline stage, particularly for high-dimensional systems or when handling matrix quantities such as Jacobians. This work introduces a novel model order reduction technique, called Discrete Compression with Sampling and Weighting (DCSW). The method combines DEIM and ECSW in a single framework, reduces the required number of DEIM modes by roughly 50%, and lowers memory use during the ECSW training offline stage by about two orders of magnitude compared to standard practice. Results across multiple test cases in solid, fluid, and thermal mechanics show that DCSW reduces extensively the offline memory usage, and provides online costs comparable to existing hyper-reduction methods while maintaining the high fidelity to the full-order model.