Reciprocal self-disclosure and need-supportive behavior are essential for close relationships, yet prior systems rarely engage the motivational underpinnings, autonomy, competence, and relatedness, that help partners internalize supportive behaviors. We introduce a Self-Determination Theory–guided chatbot that mediates self-disclosure between romantic partners by scaffolding these needs through structured questions and reflection follow-ups. In a randomized study (N=72; 36 couples), we compared three conditions: Partner Support (PS: chatbot support + partner-reflection scaffolds), Direct Support (DS: chatbot support only), and Basic Prompt (BP: questions only). PS conversations were longest and most engaged; PS and DS elicited deeper disclosures and stronger relatedness support than BP. Within PS, reflection phases concentrated partner-provided need support. Controlled motivation decreased across conditions, closeness increased only in PS, and vitality declined in BP. We contribute empirical evidence that SDT-guided mediation amplifies support and closeness, a design blueprint for relatedness technologies, and an SDT framework for advancing AI-mediated conversation design.