In spatially embedded networks such as transportation and power grids, understanding how edge removals affect connectivity is crucial for robustness analysis. This paper studies a planar graph dismantling problem under an edge-budget constraint. We propose a spanning-tree-skeleton dual-path framework that first samples multiple uniform spanning trees to capture network backbones and then adaptively selects between two complementary paths according to the budget. The small-budget path estimates a dismantlable subgraph fraction using a logarithmic density feature, while the large-budget path predicts the optimal partition count through a slope-based model. Experiments on random planar graphs demonstrate near-linear runtime scaling, consistent reductions in the largest connected component ratio, and clear budget-fragmentation trends. The method provides an interpretable and efficient approach for planar-network robustness analysis.