Given the volume of potentially false claims online, claim prioritization is essential in allocating limited human resources available for fact-checking. In this study, we perceive claim prioritization as an information retrieval (IR) task: just as multidimensional IR relevance, with many factors influencing which search results a user deems relevant, checkworthiness is also multi-faceted, subjective, and even personal, with many factors influencing how fact-checkers triage and select which claims to check. Our study investigates both the multidimensional nature of checkworthiness and effective tool support to assist fact-checkers in claim prioritization. Methodologically, we pursue Research through Design combined with mixed-method evaluation. Specifically, we develop an AI-assisted claim prioritization prototype as a probe to explore how fact-checkers use multidimensional checkworthy factors to prioritize claims, simultaneously probing fact-checker needs and exploring the design space to meet those needs. With 16 professional fact-checkers participating in our study, we uncover a hierarchical prioritization strategy fact-checkers implicitly use, revealing an underexplored aspect of their workflow, with actionable design recommendations for improving claim triage across multidimensional checkworthiness and tailoring this process with LLM integration.