Convolutional neural networks (CNNs) have emerged as a powerful tool for automatic modulation classification (AMC) by directly extracting discriminative features from raw in-phase and quadrature (I/Q) signals. However, deploying CNN-based AMC models on IoT devices remains challenging because of limited computational resources, energy constraints, and real-time processing requirements. Early-exit (EE) strategies alleviate this burden by allowing qualified samples to terminate inference at an EE branch. However, our empirical analysis reveals a critical limitation of existing confidence-based EE strategies: they predominantly select samples whose early and final predictions are correct and consistent, while failing to capture whether deeper inference can provide a tangible accuracy gain. To address this limitation, we propose BEACON, a Benefit-Aware Early-Exit framework for AMC via recoverability prediction. BEACON introduces a benefit-aware EE criterion that explicitly predicts recoverable errors, defined as instances where the final-exit branch corrects an initial early-branch misclassification. Using only short-branch observables, we design a lightweight benefit-aware predictor (LBAP) to implement this criterion, estimating the likelihood of such recoverable cases and triggering deeper inference only when an accuracy gain is expected. Extensive experiments on ResNet-18-based AMC models demonstrate that the proposed approach consistently outperforms state-of-the-art baselines, achieving a superior accuracy-computation tradeoff across diverse EE threshold settings and signal-to-noise ratio regimes. These findings validate the effectiveness of the benefit-aware criterion and its practicality for energy-efficient on-device AMC under stringent resource constraints.