Large language models (LLMs) have the potential to address social and behavioral determinants of health by transforming labor intensive workflows in resource-constrained settings. Creating LLM-based applications that serve the needs of underserved communities requires a deep understanding of their local context, but it is often the case that neither LLMs nor their developers possess this local expertise, and the experts in these communities often face severe time/resource constraints. This creates a disconnect: how can one engage in meaningful co-design of an LLM-based application for an under-resourced community when the communication channel between the LLM developer and domain expert is constrained? We explored this question through a real-world case study, in which our data science team sought to partner with social workers at a safety net hospital to build an LLM application that summarizes patients' social needs. Whereas prior works focus on the challenge of prompt tuning, we found that the most critical challenge in this setting is the careful and precise specification of \what information to surface to providers so that the LLM application is accurate, comprehensive, and verifiable. Here we present a novel co-design framework for settings with limited access to domain experts, in which the summary generation task is first decomposed into individually-optimizable attributes and then each attribute is efficiently refined and validated through a multi-tier cascading approach.