In sensitive domains, such as legal or medial domains, the correctness of information given to users is critical. To address this, the recently introduced task Conversational Tree Search (CTS) provides a graph-based framework for controllable task-oriented dialog in sensitive domains. However, a big drawback of state-of-the-art CTS agents is their long training time, which is especially problematic as a new agent must be trained every time the associated domain graph is updated. The goal of this paper is to eliminate the need for training CTS agents altogether. To achieve this, we implement a novel LLM-based method for zero-shot, controllable CTS agents. We show that these agents significantly outperform state-of-the-art CTS agents (p<0.0001; Barnard Exact test) in simulation. This generalizes to all available CTS domains. Finally, we perform user evaluation to test the agent performance in the wild, showing that our policy significantly (p<0.05; Barnard Exact) improves task-success compared to the state-of-the-art Reinforcement Learning-based CTS agent.