The minimum conflict weighted spanning tree (MCWST) problem is an \(\mathcal{N}\mathcal{P}\) -hard problem that seeks to identify a spanning tree of minimum cost subject to conflict constraints that prevent the simultaneous inclusion of certain edge pairs. The previous work in the literature addressed the conflict minimization objective using a hybrid steady-state genetic algorithm (hSSGA). Building on this work, this paper targets the subsequent objective of further minimizing the total weight of the conflict-free spanning trees generated by hSSGA. To achieve this, a variable neighborhood search (VNS) approach tailored to this weight optimization phase is proposed. The proposed VNS begins with a conflict-free solution and iteratively explores four neighborhood structures combined with local search and a reset mechanism to escape local optima. The experimental findings demonstrate that the proposed hSSGA+VNS framework reliably produces either the best-known or superior solutions, while maintaining consistent performance across various instance categories and sizes.