Timely identification of cardiac arrhythmia (abnormal heartbeats) is vital for early diagnosis of cardiovascular diseases. Wearable healthcare devices facilitate this process by recording heartbeats through electrocardiogram (ECG) signals and using AI-driven hardware to classify them into arrhythmia classes. Spiking neural networks (SNNs) are well-suited for such hardware as they consume low energy due to event-driven operation. However, their energy-efficiency and accuracy are constrained by encoding methods that translate real-valued ECG data into spikes. In this paper, we present an SNN-based ECG classification architecture featuring a new adaptive multi-threshold spike encoding scheme. This scheme adjusts encoding window and granularity based on the importance of ECG data samples, to capture essential information with fewer spikes. We develop a high-accuracy SNN model for such spike representation, by proposing a technique specifically tailored to our encoding. We design a hardware architecture for this model, which incorporates optimized layer post-processing for energy-efficient data-flow and employs fixed-point quantization for computational efficiency. Moreover, we integrate this architecture with our encoding scheme into a system-on-chip implementation using TSMC 40 nm technology. Our approach provides up to 5.1x energy-efficiency compared to state-of-the-art SNN-based ECG classifiers, with high accuracy.