Risk assessment tools in healthcare commonly employ point-based scoring systems that map patients to ordinal risk categories via thresholds. While electronic health record (EHR) data presents opportunities for data-driven optimization of these tools, two fundamental challenges impede standard supervised learning: (1) labels are often available only for extreme risk categories due to intervention-censored outcomes, and (2) misclassification cost is asymmetric and increases with ordinal distance. We propose a mixed-integer programming (MIP) framework that jointly optimizes scoring weights and category thresholds in the face of these challenges. Our approach prevents label-scarce category collapse via threshold constraints, and utilizes an asymmetric, distance-aware objective. The MIP framework supports governance constraints, including sign restrictions, sparsity, and minimal modifications to incumbent tools, ensuring practical deployability in clinical workflows. We further develop a continuous relaxation of the MIP problem to provide warm-start solutions for more efficient MIP optimization. We apply the proposed score optimization framework to a case study of inpatient falls risk assessment using the Johns Hopkins Fall Risk Assessment Tool.