Block-sparse signal recovery without knowledge of block sizes and boundaries, such as those encountered in multi-antenna mmWave channel models, is a hard problem for compressed sensing (CS) algorithms. We propose a novel Sparse Bayesian Learning (SBL) method for block-sparse recovery based on popular CS based regularizers with the function input variable related to total variation (TV). Contrary to conventional approaches that impose the regularization on the signal components, we regularize the SBL hyperparameters. This iterative TV-regularized SBL algorithm employs a majorization-minimization approach and reduces each iteration to a convex optimization problem, enabling a flexible choice of numerical solvers. The numerical results illustrate that the TV-regularized SBL algorithm is robust to the nature of the block structure and able to recover signals with both block-patterned and isolated components, proving useful for various signal recovery systems.