Large language models (LLMs) are increasingly used to simulate public opinion, yet their validity in sensitive policy domains remains underexplored. We evaluate whether LLMs can reproduce attitudes toward suicide prevention policies using 32 questions drawn from seven nationally representative U.S. surveys (2023-2025). We systematically vary demographic conditioning (race/ethnicity, gender, age, education, income, party), prompt framing (direct elicitation, respondent embodiment, specialist embodiment), and model architecture (GPT-5 Nano, DeepSeek V3.2, Meta Llama 3.1 8B, Mistral Small 24B). Across 811,560 prompts, the mean absolute error—the average gap between predicted and human response distributions—is 23 percentage points. We also find that LLM responses to demographic-conditioned prompts diverge substantially from prompts without demographic information. In short, what distribution LLMs draw on when generating responses to sensitive polling questions remains unclear. Model choice matters more than framing for accuracy, whereas refusal behavior varies sharply across models and prompt designs. Our findings highlight the limitations of LLMs for social simulation in the context of sensitive topics.