Peer-run organizations (PROs) provide critical, recovery-based behavioral health support rooted in lived experience. As large language models (LLMs) enter this domain, their scale, conversationality, and opacity introduce new challenges for situatedness, trust, and autonomy. Partnering with Collaborative Support Programs of New Jersey (CSPNJ), a statewide PRO in the Northeastern United States, we used comicboarding, a co-design method, to conduct workshops with 16 peer specialists and 10 service users exploring perceptions of integrating an LLM-based recommendation system into peer support. Findings show that depending on how LLMs are introduced, constrained, and co-used, they can reconfigure in-room dynamics by sustaining, undermining, or amplifying the relational authority that grounds peer support. We identify opportunities, risks, and mitigation strategies across three tensions: bridging scale and locality, protecting trust and relational dynamics, and preserving peer autonomy amid efficiency gains. We contribute design implications that center lived-experience-in-the-loop, reframe trust as co-constructed, and position LLMs not as clinical tools but as relational collaborators in high-stakes, community-led care.