Reliable transmission of 3D point clouds over wireless channels is challenging due to time-varying signal-to-noise ratio (SNR) and limited bandwidth. This paper introduces sensitivity-aware filtering and transmission (SAFT), a learned transmission framework that integrates a Point-BERT-inspired encoder, a sensitivity-guided token filtering (STF) unit, a quantization block, and an SNR-aware decoder for adaptive reconstruction. Specifically, the STF module assigns token-wise importance scores based on the reconstruction sensitivity of each token under channel perturbation. We further employ a training-only symbol-usage penalty to stabilize the discrete representation, without affecting the transmitted payload. Experiments on ShapeNet, ModelNet40, and 8iVFB show that SAFT improves geometric fidelity (D1/D2 PSNR) compared with a separate source--channel coding pipeline (G-PCC combined with LDPC and QAM) and existing learned baselines, with the largest gains observed in low-SNR regimes, highlighting improved robustness under limited bandwidth.