We study a resource-constrained variant of the Random Disambiguation Path (RDP) problem, a generalization of the Stochastic Obstacle Scene (SOS) problem, in which a navigating agent must reach a target in a spatial environment populated with uncertain obstacles. Each ambiguous obstacle may be disambiguated at a (possibly) heterogeneous resource cost, subject to a global disambiguation budget. We formulate this constrained planning problem as a Weight-Constrained Shortest Path Problem (WCSPP) with risk-adjusted edge costs that incorporate probabilistic blockage and traversal penalties. To solve it, we propose a novel algorithmic framework-COLOGR-combining Lagrangian relaxation with a two-phase vertex elimination (TPVE) procedure. The method prunes infeasible and suboptimal paths while provably preserving the optimal solution, and leverages dual bounds to guide efficient search. We establish correctness, feasibility guarantees, and surrogate optimality under mild assumptions. Our analysis also demonstrates that COLOGR frequently achieves zero duality gap and offers improved computational complexity over prior constrained path-planning methods. Extensive simulation experiments validate the algorithm's robustness across varying obstacle densities, sensor accuracies, and risk models, consistently outperforming greedy baselines and approaching offline-optimal benchmarks. The proposed framework is broadly applicable to stochastic network design, mobility planning, and constrained decision-making under uncertainty.