Given the massive volume of potentially false claims circulating 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. We develop an AI-assisted claim prioritization prototype as a probe to explore how fact-checkers use multidimensional checkworthiness factors in claim prioritization, simultaneously probing fact-checker needs while also exploring the design space to meet those needs. Our study with 16 professional fact-checkers investigates: 1) how participants assessed the relative importance of different checkworthy dimensions and apply different priorities in claim selection; 2) how they created customized GPT-based search filters and the corresponding benefits and limitations; and 3) their overall user experiences with our prototype. Our work makes a conceptual contribution between multidimensional IR relevance and fact-checking checkworthiness, with findings demonstrating the value of corresponding tooling support. Specifically, we uncovered a hierarchical prioritization strategy fact-checkers implicitly use, revealing an underexplored aspect of their workflow, with actionable design recommendations for improving claim triage across multi-dimensional checkworthiness and tailoring this process with LLM integration.