Model Order Reduction (MOR) is a fundamental technology for creating comprehensive Digital Twins (DTs) in science and engineering by reducing the computational load of high-fidelity models. However, while effective for linear systems, Physics-Based MOR methods face challenges with parametric and nonlinear settings. Hyper-Reduction (HR) techniques can address these issues but are often intrusive, as they require access to full-model operators. On the other hand, purely data-driven methods are non-intrusive but often fail to maintain key physical properties, leading to instability in Reduced Order Models (ROMs). This work proposes the use of a non-intrusive Physics-Augmented Neural Network approach, which leverages the Partially Input Convex Neural Network (PICNN) architecture. This approach allows to create a parametric non-intrusive hyper-reduced model for nonlinear structural elastic finite element formulations. The use of a PICNN ensures the a-priori enforcement of key physical properties such as hyper-elasticity, material stability, and material consistency. To showcase the non-intrusive properties, the proposed approach is applied to data stemming from a commercial nonlinear structural finite element solver. The results validate the non-intrusive properties of the proposed approach and highlight the need for physics-augmentation, which inherently embeds essential physical constraints to ensure stable and robust behavior. Furthermore, the use of the proposed approach with a commercial solver demonstrates its potential use and effectiveness for industrial DT applications.