refactor(backend): clean up router registration pattern

Issue #6: Multiple routers per file made main.py verbose.

Changes:
- Add 'routers' list export to booking.py and invites.py
- Update main.py to iterate over router lists for multi-router modules
- Keep explicit registration for single-router modules
- Cleaner separation between simple and complex route modules

backend/main.py

@@ -42,16 +42,17 @@ app.add_middleware(
     allow_credentials=True,
 )
 
-# Include routers
+# Include routers - modules with single router
 app.include_router(auth_routes.router)
 app.include_router(sum_routes.router)
 app.include_router(counter_routes.router)
 app.include_router(audit_routes.router)
 app.include_router(profile_routes.router)
-app.include_router(invites_routes.router)
-app.include_router(invites_routes.admin_router)
 app.include_router(availability_routes.router)
-app.include_router(booking_routes.router)
-app.include_router(booking_routes.appointments_router)
-app.include_router(booking_routes.admin_appointments_router)
 app.include_router(meta_routes.router)
+
+# Include routers - modules with multiple routers
+for r in invites_routes.routers:
+    app.include_router(r)
+for r in booking_routes.routers:
+    app.include_router(r)

backend/routes/booking.py

@@ -377,3 +377,7 @@ async def admin_cancel_appointment(
     await db.refresh(appointment)
 
     return _to_appointment_response(appointment)  # Uses eager-loaded relationship
+
+
+# All routers from this module for easy registration
+routers = [router, appointments_router, admin_appointments_router]

backend/routes/invites.py

@@ -231,3 +231,7 @@ async def revoke_invite(
     await db.refresh(invite)
 
     return build_invite_response(invite)
+
+
+# All routers from this module for easy registration
+routers = [router, admin_router]