Neuromorphic computing, which is inspired by the dynamics of biological nervous systems, has emerged as a novel computing paradigm capable of efficiently performing not only neural networks for cognitive tasks but also non-cognitive tasks such as sparse modeling and graph search. Spiking Locally Competitive Algorithm (S-LCA) is an efficient LASSO solver with order-of-magnitude advantages in terms of power consumption and latency. However, conventional S-LCA requires a significant number of timesteps to converge, making it difficult to connect with the latest spiking neural networks for image recognition that can be performed in few timesteps or difficult to use as preprocessing for event-based data. Therefore, we propose an S-LCA trained by Backpropagation Through Time for ultra-low latency LASSO. In our proposed model, L2 Norm and Batch Normalization are used to improve accuracy. Through validation using image datasets, we achieved a significant reduction in timesteps, demonstrating the potential for fast operation and low energy consumption. Moreover, the results show that the proposed algorithm achieves accuracy comparable to non-spiking analog LCA.