Caregivers seeking AI-mediated support express complex needs -- information-seeking, emotional validation, and distress cues -- that warrant careful evaluation of response safety and appropriateness. Existing AI evaluation frameworks, primarily focused on general risks (toxicity, hallucinations, policy violations, etc), may not adequately capture the nuanced risks of LLM-responses in caregiving-contexts. We introduce RubRIX (Rubric-based Risk Index), a theory-driven, clinician-validated framework for evaluating risks in LLM caregiving responses. Grounded in the Elements of an Ethic of Care, RubRIX operationalizes five empirically-derived risk dimensions: Inattention, Bias & Stigma, Information Inaccuracy, Uncritical Affirmation, and Epistemic Arrogance. We evaluate six state-of-the-art LLMs on over 20,000 caregiver queries from Reddit and ALZConnected. Rubric-guided refinement consistently reduced risk-components by 45-98% after one iteration across models. This work contributes a methodological approach for developing domain-sensitive, user-centered evaluation frameworks for high-burden contexts. Our findings highlight the importance of domain-sensitive, interactional risk evaluation for the responsible deployment of LLMs in caregiving support contexts. We release benchmark datasets to enable future research on contextual risk evaluation in AI-mediated support.