Regularized Black Hole Solution from a New String Cloud Source

We construct a new family of regular black hole solutions supported by the novel Letelier-Alencar string cloud and regularized through a rational Dagum-type distribution. The regulator smooths the matter profile and ensures finite curvature invariants, yielding a geometry that interpolates between a string-cloud exterior and an anti--de Sitter core. We analyze the energy conditions, identifying where the null, weak, dominant and strong conditions hold or fail across the core and exterior. The parameter space for horizon formation is mapped and the thermodynamic propertie -- mass, Hawking temperature, entropy and heat capacity -- are derived; notably, the entropy depends only on the regularization scale while the string parameter modifies temperature and heat capacity. Employing R\'enyi non-extensive entropy and the topological thermodynamics approach, we show the non-extensive deformation stabilizes the system and removes the standard phase transition. Finally, we compute the shadow radius and derive constraints compatible with current Event Horizon Telescope bounds for Sgr~A* and M87*.