As live streaming services grow, many platforms offer short videos and live streams to meet diverse needs. Short videos carry substantial traffic and rich behavior signals, whereas live streaming is a core conversion scenario with sparse behavior data, making cold start severe. Transferring user interests from short videos to live streaming recommendation can alleviate these issues. Meanwhile, short videos and live streams are complex multimodal items, and integrating multimodal signals improves recommendation performance. Although Multimodal Large Language Models (MLLMs) show strong multimodal understanding and reasoning, their application to cross-domain recommendation remains underexplored. To this end, we propose Reasoning-Guided Cross-Domain Representation Learning (RGCD-Rep), a reasoning-guided framework for cross-domain recommendation from short videos to live streams. RGCD-Rep introduces MLLM reasoning resource-efficiently and learns transferable item representations guided by behavioral collaboration via two-stage training. First, reasoning-aware distillation lets a frozen teacher MLLM generate structured cross-domain reasoning knowledge and distills it into a lightweight student MLLM. Second, transferability-guided cross-domain representation learning decomposes item representations into transferable and domain residual representations. The resulting representations are computed offline and integrated into downstream retrieval tasks, enabling low-cost industrial deployment. Extensive offline experiments demonstrate RGCD-Rep's superiority. After deployment in Kuaishou's live streaming recommendation system, A/B tests show significant gains across multiple core business metrics, confirming its effectiveness and practicality in real industrial scenarios. RGCD-Rep is fully deployed and serves over 400 million users daily.