feat: build the first end-to-end agent chat loop

[ibourgeois]

Summary

Implement the first usable private chat flow where a user can send a message to an agent and receive a persisted response.

Why This Matters

This is the first meaningful product goal: using Katra to actually talk to an agent.

Acceptance Criteria

• a user can open a private chat with an agent participant
• a user message is persisted before agent execution starts
• the agent response is persisted back into the same chat
• the chat transcript reloads cleanly from storage
• the implementation is intentionally narrow and can use one simple agent path before broader orchestration exists

Notes

This issue should stay focused on the first happy-path conversation loop. Advanced orchestration, background workers, and room-specific tooling can come later.