Multi-Agent Path Finding (MAPF) algorithms, including those for car-like robots and grid-based scenarios, face significant computational challenges due to expensive heuristic calculations. Traditional heuristic caching assumes that the heuristic function depends only on the state, which is incorrect in constraint-based search algorithms (e.g., CBS, MAPF-LNS, MAP2) where constraints from conflict resolution make the search space context-dependent. We propose \textbf{CAHC} (Conflict-Aware Heuristic Caching), a general framework that caches heuristic values based on both state and relevant constraint context, addressing this fundamental limitation. We demonstrate CAHC through a case study on CL-CBS for car-like robots, where we combine conflict-aware caching with an adaptive hybrid heuristic in \textbf{CAR-CHASE} (Car-Like Robot Conflict-Aware Heuristic Adaptive Search Enhancement). Our key innovations are (1) a compact \emph{conflict fingerprint} that efficiently encodes which constraints affect a state's heuristic, (2) a domain-adaptable relevance filter using spatial, temporal, and geometric criteria, and (3) a modular architecture that enables systematic application to diverse MAPF algorithms. Experimental evaluation on 480 CL-CBS benchmark instances demonstrates a geometric mean speedup of 2.46$\times$ while maintaining solution optimality. The optimizations improve success rate from 77.9\% to 84.8\% (+6.9 percentage points), reduce total runtime by 70.1\%, and enable solving 33 additional instances. The framework's general architecture makes it applicable as a reliable optimization technique for MAP2, MAPF-LNS, and other constraint-based MAPF algorithms.