Diabetic retinopathy (DR) is a leading cause of permanent vision loss worldwide. It refers to irreversible retinal damage caused due to elevated glucose levels and blood pressure. Regular screening for DR can facilitate its early detection and timely treatment. Neural network-based DR classifiers can be leveraged to achieve such screening in a convenient and automated manner. However, these classifiers suffer from reliability issue where they exhibit strong performance during development but degraded performance after deployment. Moreover, they do not provide supplementary information about the prediction outcome, which severely limits their widespread adoption. Furthermore, energy-efficient deployment of these classifiers on edge devices remains unaddressed, which is crucial to enhance their global accessibility. In this paper, we present a reliable and energy-efficient hardware for DR detection, suitable for deployment on edge devices. We first develop a DR classification model using custom training data that incorporates diverse image quality and image sources along with improved class balance. This enables our model to effectively handle both on-field variations in retinal images and minority DR classes, enhancing its post-deployment reliability. We then propose a pseudo-binary classification scheme to further improve the model performance and provide supplementary information about the model prediction. Additionally, we present an energy-efficient hardware design for our model using memristor-based computation-in-memory, to facilitate its deployment on edge devices. Our proposed approach achieves reliable DR classification with three orders of magnitude reduction in energy consumption over state-of-the-art hardware platforms.