The recent growth of on-device Large Language Model (LLM) inference has driven significant interest in device-edge collaborative LLM inference. As a promising architecture, Speculative Decoding (SD) is increasingly adopted where a lightweight draft model rapidly generates candidate tokens to be verified by a powerful target model. However, a fundamental challenge lies in achieving per-token resource scheduling to effectively adapt SD paradigm to resource-constrained edge environment. This paper proposes a Generative Entropy- and Lyapunov-based Adaptive Token Offloading framework, named GELATO, to maximize decoding throughput under energy constraints in a device-edge collaborative SD system. Specifically, an outer drift-plus-penalty loop makes online decisions to establish a reference drafting budget, managing long-term energy-throughput trade-off. Further, a nested entropy-driven generation mechanism executes early exiting to adapt to per-token dynamic generative uncertainty. Theoretical analysis establishes a rigorous performance bound on long-term throughput for GELATO. Extensive evaluations demonstrate that GELATO achieves a globally optimal tradeoff, outperforming state-of-the-art distributed SD architectures by 64.98% in token throughput and reducing energy consumption by 47.47% under resource-constrained environments, while preserving LLM decoding quality.