Recent advances in AI and machine learning (ML) promise significant transformations in the future delivery of healthcare. Despite a surge in research and development, few works have moved beyond demonstrations of technical feasibility and algorithmic performance. However, to realize many of the ambitious visions for how AI can contribute to clinical impact requires the closer design and study of AI tools or interventions within specific health and care contexts. This article outlines our collaborative, human-centered approach to developing an AI application that predicts treatment outcomes for patients who are receiving human-supported, internet-delivered Cognitive Behavioral Therapy (iCBT) for symptoms of depression and anxiety. Intersecting the fields of HCI, AI, and healthcare, we describe how we addressed the specific challenges of (1) identifying clinically relevant AI applications ; and (2) designing AI applications for sensitive use contexts like mental health. Aiming to better assist the work practices of iCBT supporters, we share how learnings from an interview study with 15 iCBT supporters surfaced their practices and information needs and revealed new opportunities for the use of AI. Combined with insights from the clinical literature and technical feasibility constraints, this led to the development of two clinical outcome prediction models. To clarify their potential utility for use in practice, we conducted 13 design sessions with iCBT supporters that utilized interface mock-ups to concretize the AI output and derive additional design requirements. Our findings demonstrate how design choices can impact interpretations of the AI predictions as well as supporter motivation and sense of agency. We detail how this analysis and the design principles derived from it enabled the integration of the prediction models into a production interface. Reporting on identified risks of over-reliance on AI outputs and needs for balanced information assessment and preservation of a focus on individualized care, we discuss and reflect on what constitutes a responsible, human-centered approach to AI design in this healthcare context.