This paper introduces an approach that enhances the state estimator for high-speed autonomous race cars, addressing challenges from unreliable measurements, localization failures, and computing resource management. The proposed robust localization system utilizes a Bayesian-based probabilistic approach to evaluate multimodal measurements, ensuring the use of credible data for accurate and reliable localization, even in harsh racing conditions. To tackle potential localization failures, we present a resilient navigation system which enables the race car to continue track-following by leveraging direct perception information in planning and execution, ensuring continuous performance despite localization disruptions. In addition, efficient computing is critical to avoid overload and system failure. Hence, we optimize computing resources using an efficient LiDAR-based state estimation method. Leveraging CUDA programming and GPU acceleration, we perform nearest points search and covariance computation efficiently, overcoming CPU bottlenecks. Simulation and real-world tests validate the system's performance and resilience. The proposed approach successfully recovers from failures, effectively preventing accidents and ensuring safety of the car.