Phase 2: private rooms, room roles, and invites (backend only)

Adds room_invites table + migration, owner/admin/member role enforcement
(require_room_role), and endpoints for private room creation, room
management (update/delete/leave/transfer-ownership/change-role/remove-member),
and the invite lifecycle (create/list/accept/decline/revoke). Registration
stays invite-only via the CLI from Phase 1 — this is a separate, room-level
invite system for adding existing users to private rooms.

Frontend is untouched: the UI redesign is happening separately, so this phase
is backend + tests only (35 passing). Verified no regressions in the Phase 1
open-room/WebSocket flow via manual smoke test.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

backend/app/main.py

@@ -2,7 +2,7 @@ from fastapi import FastAPI
 from starlette.middleware.sessions import SessionMiddleware
 
 from app.config import settings
-from app.routers import auth, health, rooms
+from app.routers import auth, health, invites, rooms
 from app.ws.chat import router as ws_router
 from app.ws.connection_manager import ConnectionManager
 
@@ -23,6 +23,7 @@ def create_app() -> FastAPI:
     app.include_router(health.router)
     app.include_router(auth.router)
     app.include_router(rooms.router)
+    app.include_router(invites.router)
     app.include_router(ws_router)
 
     return app