Deep Operator Networks (DeepONets) and their physics-informed variants have shown significant promise in learning mappings between function spaces of partial differential equations, enhancing the generalization of traditional neural networks. However, for highly nonlinear real-world applications like aerospace composites processing, existing models often fail to capture underlying solutions accurately and are typically limited to single input functions, constraining rapid process design development. This paper introduces an advanced physics-informed DeepONet tailored for such complex systems with multiple input functions. Equipped with architectural enhancements like nonlinear decoders and effective training strategies such as curriculum learning and domain decomposition, the proposed model handles high-dimensional design spaces with significantly improved accuracy, outperforming the vanilla physics-informed DeepONet by two orders of magnitude. Its zero-shot prediction capability across a broad design space makes it a powerful tool for accelerating composites process design and optimization, with potential applications in other engineering fields characterized by strong nonlinearity.