There has been growing interest in computing rate-distortion functions for real-world data, as they can provide a theoretical benchmark for compression problems. However, a generalized form of rate-distortion that includes side information and coding for computing has been underexplored, despite its relevance in modern compression problems. To address this gap, we propose a new method for estimating the rate-distortion function for computing with side information, using a Lagrangian framework with neural network-parametrized encoding and decoding strategies. This approach enables targeting specific points on the rate-distortion curve through gradient-based optimization. Our methodology is validated in synthetic environments where rate-distortion functions are known, ensuring accuracy in estimation. Additionally, we extend its application to practical, high-dimensional channel state information compression scenarios. We provide rate-distortion estimation results on these scenarios, which in turn enables us to quantify the usefulness of side information in the practical scenarios.