Reducing energy consumption in the building sector has driven the search for more sustainable construction methods. This study explores the potential of cork-modified mortars reinforced with basalt fabric, focusing on optimizing both mechanical and hygroscopic properties. Six mortar mixtures were produced using a breathable structural mortar made from pure natural hydraulic lime, incorporating varying percentages (0–3%) of cork granules (Quercus suber) as lightweight aggregates. Micro-computed tomography was first used to assess the homogeneity of the mixtures, followed by flow tests to evaluate workability. The mixtures were then tested for water absorption, compressive strength, and adhesion to tuff and clay brick surfaces. Adhesion was measured through pull-off tests, to evaluate internal bonding strength. Additionally, this study examined the relationship between surface roughness and bond strength in FRLM composites, revealing that rougher surfaces significantly improved adhesion to clay and tuff bricks. These findings suggest that cork-reinforced mortars offer promising potential for sustainable construction, achieving improved hygroscopic performance, sufficient mechanical strength, internal bonding, and optimized surface adhesion.