Autonomous drone pursuit requires not only detecting drones but also predicting their trajectories in a manner that enables kinematically feasible interception. Existing tracking methods optimize for prediction accuracy but ignore pursuit feasibility, resulting in trajectories that are physically impossible to intercept 99.9% of the time. We propose Perception-to-Pursuit (P2P), a track-centric temporal reasoning framework that bridges detection and actionable pursuit planning. Our method represents drone motion as compact 8-dimensional tokens capturing velocity, acceleration, scale, and smoothness, enabling a 12-frame causal transformer to reason about future behavior. We introduce the Intercept Success Rate (ISR) metric to measure pursuit feasibility under realistic interceptor constraints. Evaluated on the Anti-UAV-RGBT dataset with 226 real drone sequences, P2P achieves 28.12 pixel average displacement error and 0.597 ISR, representing a 77% improvement in trajectory prediction and 597x improvement in pursuit feasibility over tracking-only baselines, while maintaining perfect drone classification accuracy (100%). Our work demonstrates that temporal reasoning over motion patterns enables both accurate prediction and actionable pursuit planning.