We introduce Paired-CSLiDAR (CSLiDAR), a cross-source aerial-ground LiDAR benchmark for single-scan pose refinement: refining a ground-scan pose within a 50 m-radius aerial crop. The benchmark contains 12,683 ground-aerial pairs across 6 evaluation sites and per-scan reference 6-DoF alignments for sub-meter root-mean-square error (RMSE) evaluation. Because aerial scans capture rooftops and canopy while ground scans capture facades and under-canopy, the two modalities share only a fraction of their geometry, primarily the terrain surface, causing standard registration methods and learned correspondence models to converge to metrically incorrect local minima. We propose Residual-Guided Stratified Registration (RGSR), a training-free, geometry-only refinement pipeline that exploits the shared ground plane through height-stratified ICP, reversed registration directions, and confidence-gated accept-if-better selection. RGSR achieves 86.0% S@0.75 m and 99.8% S@1.0 m on the primary benchmark of 9,012 scans, outperforming both the confidence-gated cascade at 83.7% and GeoTransformer at 76.3%. We validate RMSE-based pose selection with independent survey control and trajectory consistency, and show that added Fourier-Mellin BEV proposals can reduce RMSE while increasing actual pose error under extreme partial overlap. The dataset and code are being prepared for public release.