Trajectory prediction is a fundamental task for autonomous systems, requiring complex reasoning about multi-agent interactions and intents. Large language models (LLMs) have recently been adopted for this task, as they provide strong contextual reasoning and interpretable, language-based trajectory representations. However, these LLM-based predictors are extremely memory- and compute-intensive, making them difficult to deploy on resource-constrained edge devices such as on-board computers in autonomous robots. To bridge this gap, we propose BitTP, which converts an LLM-based trajectory predictor into a lightweight bitlinear architecture. We demonstrate that weight-only quantization to 1.58-bit (BitTP-Weight) is optimal. Crucially, activations must remain in full precision, as quantizing them leads to severe degradation and instability in spatio-temporal reasoning. Empirically, BitTP-Weight not only preserves but improves prediction quality over the full-precision (BF16) LLM baseline, reducing ADE by 14.29% and FDE by 20.97% on average, while simultaneously reducing memory usage and inference latency relative to other quantization methods. These results demonstrate that carefully designed quantization acts as an effective regularizer, enabling the practical deployment of sophisticated LLM-based reasoning on edge devices. Code is available at: https://github.com/MintCat98/BitTP.