Refactor AgentAPI to use the claude-code SDK when applicable

[johnstcn]

Problem

We currently use a CLI-agnostic approach for communicating with Claude Code.
This involves spawning a virtual terminal, sending keystrokes over a PTY, and scraping the output.
This mostly works 'fine' but occasionally can result in rendering issues or garbled output.

Proposed Solution

Update AgentTypeClaudeCode to use the Claude Code SDK. This is the same underlying implementation used by the Python SDK.

Alternatively, if a pure-Go SDK becomes available, we could investigate that first.

[bpmct]

Huge fan of us supporting native SDKs for a better UX when possible!