As the development of lightweight deep learning algorithms, various deep neural network (DNN) models have been proposed for the remote sensing scene classification (RSSC) application. However, it is still challenging for these RSSC models to achieve optimal performance among model accuracy, inference latency, and energy consumption on resource-constrained edge devices. In this paper, we propose a lightweight RSSC framework, which includes a distilled global filter network (GFNet) model and an early-exit mechanism designed for edge devices to achieve state-of-the-art performance. Specifically, we first apply frequency domain distillation on the GFNet model to reduce model size. Then we design a dynamic early-exit model tailored for DNN models on edge devices to further improve model inference efficiency. We evaluate our E3C model on three edge devices across four datasets. Extensive experimental results show that it achieves an average of 1.3x speedup on model inference and over 40% improvement on energy efficiency, while maintaining high classification accuracy.