Brain-computer interfaces are built upon precise decoding of neural signals. Spike sorting is a crucial step in this process to extract individual neural activities from complex neural data. This paper presents a spiking neural network (SNN) framework for efficient spike sorting, which is named SIFT-RSNN. In this new framework, raw neural signals are encoded into spike trains using a threshold-based temporal encoding strategy, where continuous time signals are converted into spikes only when the signal's variation crosses a defined threshold. Misfiring spikes are refined using a sparse-integrated filtering module, which enhances data sparsity for pattern learning. Overall, the SIFT-RSNN with a membrane shortcut structure has efficient feature transfer and improved generalization performance. When tested on the Difficult1 and Difficult2 subsets of Leicester datasets, the SIFT-RSNN scored higher than current state-of-the-art methods at 96.2% and 99.6%, respectively. The authors conducted it on a compute-in-memory platform with 8kmemristor cells, using quantization-free mapping method, and propose two strategies to mitigate non-ideal hardware effects. After optimization, the algorithm still outperforms existing spike sorting methods and has higher robustness. The memristor platform only exhibits a 2% and 1.5% accuracy drop compared to software results on the two difficult subsets. Additionally, it achieves 3.52 μJ energy consumption and 0.5 ms latency per inference.