Language models are increasingly being deployed for conversational support in informal caregiving contexts, where interactions often extend beyond information-seeking: caregivers seek emotional reassurance, guidance, and help, while navigating uncertain, relationally complex care decisions. Yet most safety evaluations assess model behavior under generic prompts, leaving a critical question unexamined: does a model's safety profile change with its support role? We study this by operationalizing four expert-reviewed support roles grounded in social support theory: Inform, Coach, Relate, and Listen, and comparing them against two baseline controls: a basic prompting condition and a retrieval-augmented generation (RAG) condition. We evaluate across three language models (GPT-4o-mini, Llama-3.1-8B-Instruct, and MedGemma-1.5-4b-it) on 5,000 real-world queries from online Alzheimer's Disease and Related Dementias (ADRD) communities. We find that the LLM's support role systematically shapes both the prevalence and composition of interactional risks. Furthermore, a human evaluation study reveals a perceived quality--safety tension: more directive, information-oriented roles are rated as more helpful and trustworthy despite exhibiting elevated interactional risk profiles. We release ~90,000 support role-conditioned model responses with risk annotations as an ecologically grounded resource for research on safer LLM-mediated conversational support.