Texture segmentation still constitutes an ongoing challenge, especially when processing large-size images. Recently, procedures integrating a scale-free (or fractal) wavelet-leader model allowed the problem to be reformulated in a convex optimization framework by including a TV penalization. In this case, the TV penalty plays a prominent role with respect to the data fidelity term, which makes the approach costly in terms of memory and computation cost. The present contribution aims to investigate the potential of recent block-coordinate dual and primal-dual proximal algorithms for overcoming this numerical issue. Our study shows that a key ingredient in the success of the proposed block-coordinate approaches lies in the design of the blocks of variables which are updated at each iteration. Numerical experiments conducted over synthetic textures having piece-wise constant fractal properties confirm our theoretical analysis. The proposed lattice block design strategy is shown to yield significantly lower memory and computational requirements.