fix(api): register the AI chat router once

[CoreyLeath-code]

Problem

backend/app/main.py includes the same AI chat router twice at /api/chat. This produces duplicate route registration and makes the public API contract harder to reason about.

Proposed resolution

• Retain a single router registration.
• Add a regression test that asserts exactly one POST /api/chat/ route is exposed.

Acceptance criteria

• The application exposes one POST chat route.
• Existing AI endpoint tests remain green.
• No behavior or route path changes for clients.