TRCLI-213 Refactor core api handler class into separate handlers to improve code quality

[acuanico-tr-galt]

• Single Responsibility - Extracted common operations (parsing, deduplication, validation) into focused utility functions so each function and handler has only one reason to change. God Class Extraction: Broke down ApiRequestHandler from 2,384 lines with 10+ responsibilities into 9 focused handlers (LabelManager, ReferenceManager, CaseHandler, etc.), reducing it by
  53.7% to 1,103 lines.
• Open/Closed - In the merge_references (as example) function accepts strategy parameter for new behaviours without modifying underlying code. We also used delegation pattern to maintain backward compatibility.
• Interface Segregation - We broke down the god classes (e.g. api_request_handler) into small, focused utility functions, so handlers import only what they need.
• Dependency Inversion - Handlers now depend on abstract types, and utility interfaces rather than concrete implementations. We also used callback injection pattern to share functionality between handlers without tight coupling.

Potential impacts

None. All refactors in this PR are all backwards compatible.

PR Tasks

• Unit tests added/updated