We present a case study of Persona-L, a system that leverages large language models (LLMs) and retrieval-augmented generation (RAG) to model personas of people with Down syndrome. Existing approaches to persona creation can often lead to oversimplified or stereotypical profiles of people with Down syndrome. To that end, we built stereotype detection capabilities into Persona-L as a design probe to open conversations. We then conducted interviews with caregivers and healthcare professionals (N=10) to examine how Down syndrome stereotypes could manifest in the content and delivery of LLM outputs, and interface design. Our findings show the challenges in defining stereotypes, and reveal potential pathways where stereotypes could emerge, including the training data, LLM outputs and interface design. This highlights the need for participatory methods that capture the heterogeneity of lived experiences of people with Down syndrome.