This paper introduces FWL-ToPM and FWLAug to enable real-time full-waveform LiDAR (FWL) perception with Transformers. FWL captures complete return waveforms as histograms, preserving multi-path echoes and weak signals critical for adverse conditions, but Transformer processing suffers from redundant tokens in low-intensity regions and limited annotated data. FWL-ToPM exploits the physical intensity distribution to prune and merge tokens, selectively retaining critical waveform content while reducing computational load. Evaluated on the Ghost-FWL benchmark, it achieves over 8× speedup at real-time throughput and a nearly 9-point F1 improvement over the prior state of the art, establishing the first real-time FWL Transformer that simultaneously boosts accuracy. FWLAug complements this by augmenting training data through a physics-aware framework that preserves Time-of-Flight waveform structure per ray and frustum geometry, acting as regularization to improve generalization. It further enhances performance on both the standard benchmark and an out-of-distribution test set of unseen scenes. Code and the OOD dataset are publicly available.