The objective of this study is to develop a domain decomposition framework for handling large-scale geometries, focusing on predicting required system responses of 3D engineering components under varying test conditions. Our approach utilizes a neural operator-based framework, specifically the Deep Operator Network (DeepONet) as a representative operator model for learning the mapping between infinite-dimensional function spaces. The training dataset is constructed through a Design of Experiments (DOE), systematically varying multiple design parameters to capture the responses across different design configurations. The operator network is designed to handle unseen DOE test conditions, with the design parameters serving as inputs to the branch network. To address the computational challenges associated with large-scale geometries, we implement domain decomposition, dividing the domain into smaller sub-sections for efficient operator training. Subsequently, we integrate these individual components to generate predictions for the complete design. Additionally, we incorporate boundary detection techniques using point-cloud representations to enhance accuracy at interface boundaries. Our results indicate that the operator network demonstrates robust accuracy across diverse DOE conditions and geometric configurations. Notably, higher error values are observed in extreme test cases of the DOE. Therefore, we also present results from a comprehensive cross-validation study to assess the extrapolation capability of the DeepONet framework. In conclusion, our domain decomposition-based neural operator framework, serves as an efficient data-driven solution for predicting responses to various design configurations in realistic, large-scale applications, yielding acceptable error values.