The rapid development of the Internet of Things (IoT) applications necessitates resource-efficient computing paradigms that can unify heterogeneous sensing modalities. Spiking Neural Networks (SNNs) meet this need with their event-driven and energy-efficient processing nature. However, deploying SNNs on mobile and embedded platforms is hindered by strict and fluctuating memory budgets. While prior work explores lightweight model design and system-level memory management, these methods either sacrifice accuracy or incur high runtime overhead due to timestep-dependent dynamics. To tackle these challenges, we propose a memory-adaptive framework MASI that enables efficient on-device SNN inference by combining (1) a fine-grained memory-adaptive layer slicing strategy, (2) a timestep-agnostic scheduler that maximizes memory utilization with minimal fragmentation, and (3) a timestep-aware early-exit mechanism that reduces redundant calculations. Evaluated on diverse workloads and edge devices, MASI can dynamically adapt to runtime memory availability, approximately reducing memory usage by 20.67% and inference latency by 58.53% on average with negligible accuracy loss compared to other feasible on-device implementations under memory constraints.