Chinese paper-cutting, an Intangible Cultural Heritage (ICH), faces challenges from the erosion of traditional culture due to the prevalence of realism alongside limited public access to cultural elements. While generative AI can enhance paper-cutting design with its extensive knowledge base and efficient production capabilities, it often struggles to align content with cultural meaning due to users' and models' lack of comprehensive paper-cutting knowledge. To address these issues, we conducted a formative study (N=7) to identify the workflow and design space, including four core factors (Function, Subject Matter, Style, and Method of Expression) and a key element (Pattern). We then developed HarmonyCut, a generative AI-based tool that translates abstract intentions into creative and structured ideas. This tool facilitates the exploration of suggested related content (knowledge, works, and patterns), enabling users to select, combine, and adjust elements for creative paper-cutting design. A user study (N=16) and an expert evaluation (N=3) demonstrated that HarmonyCut effectively provided relevant knowledge, aiding the ideation of diverse paper-cutting designs and maintaining design quality within the design space to ensure alignment between form and cultural connotation.