In AI-powered e-commerce livestreaming, digital avatars require real-time responses to drive engagement, a task for which high-latency Large Reasoning Models (LRMs) are ill-suited. We introduce LiveThinking, a practical two-stage optimization framework to bridge this gap. First, we address computational cost by distilling a 670B teacher LRM into a lightweight 30B Mixture-of-Experts (MoE) model (3B active) using Rejection Sampling Fine-Tuning (RFT). This reduces deployment overhead but preserves the teacher's verbose reasoning, causing latency. To solve this, our second stage employs reinforcement learning with Group Relative Policy Optimization (GRPO) to compress the model's reasoning path, guided by a multi-objective reward function balancing correctness, helpfulness, and brevity. LiveThinking achieves a 30-fold reduction in computational cost, enabling sub-second latency. In real-world application on Taobao Live, it improved response correctness by 3.3% and helpfulness by 21.8%. Tested by hundreds of thousands of viewers, our system led to a statistically significant increase in Gross Merchandise Volume (GMV), demonstrating its effectiveness in enhancing user experience and commercial performance in live, interactive settings.