This paper considers clustered multi-task compressive sensing, a hierarchical model that solves multiple compressive sensing tasks by finding clusters of tasks that leverage shared information to mutually improve signal reconstruction. The existing inference algorithm for this model is computationally expensive and does not scale well in high dimensions. The main bottleneck involves repeated matrix inversion and log-determinant computation for multiple large covariance matrices. We propose a new algorithm that substantially accelerates model inference by avoiding the need to explicitly compute these covariance matrices. Our approach combines Monte Carlo sampling with iterative linear solvers. Our experiments reveal that compared to the existing baseline, our algorithm can be up to thousands of times faster and an order of magnitude more memory-efficient.