FastAPI 0.137 stopped flattening include_router into app.routes — each include is
now an _IncludedRouter (a BaseRoute with no .path), so `{r.path for r in
app.routes}` raised AttributeError and the two router-registration tests failed
(the bump arrived via the claude-agent-sdk update in uv.lock). Add
_registered_paths(): OpenAPI schema paths (the stable public contract) plus each
included router's prefix, which also covers the websocket /ws mount (never in the
schema). Drops the now-incorrect type: ignore[attr-defined].