Facilitating self-disclosure without causing discomfort remains a difficult task—especially for AI systems. In real-world applications such as career counseling, wellbeing support, and onboarding interviews, eliciting personal information like concerns, goals, and personality traits is essential. However, asking such questions directly often leads to discomfort and disengagement. We address this issue with RaPSIL (Rapport-aware Preference-guided Self-disclosure Interview Learner), a two-stage LLM-based system that fosters natural, engaging conversations to promote self-disclosure. In the first stage, RaPSIL selectively imitates interviewer utterances that have been evaluated by LLMs for both strategic effectiveness and social sensitivity. It leverages LLMs as multi-perspective judges in this selection process. In the second stage, it conducts self-play simulations, using the Reflexion framework to analyze failures and expand a database with both successful and problematic utterances. This dual learning process allows RaPSIL to go beyond simple imitation, improving its ability to handle sensitive topics naturally by learning from both successful and failed utterances. In a comprehensive evaluation with real users, RaPSIL outperformed baselines in enjoyability, warmth, and willingness to re-engage, while also capturing self-descriptions more accurately. Notably, its impression scores remained stable even during prolonged interactions, demonstrating its ability to balance rapport building with effective information elicitation. These results show that RaPSIL enables socially aware AI interviewers capable of eliciting sensitive personal information while maintaining user trust and comfort—an essential capability for real-world dialogue systems.