Route LM Studio requests through the shared HTTP client (#34678)

## What changed

- Use the configured route-aware HTTP client pool for LM Studio server requests.
- Add connection-timeout support to `HttpClientBuilder` and route-aware pools, and keep LM Studio's five-second limit scoped to connection establishment.
- Verify that LM Studio accepts a response that arrives after the connection timeout has elapsed once the connection is established.

GitOrigin-RevId: c4300f4b5d37c4418822783ab09cb50d506ee423

codex-rs/deny.toml

@@ -243,7 +243,6 @@ deny = [
         "codex-app-server",
         "codex-core",
         "codex-exec-server",
-        "codex-lmstudio",
         "codex-ollama",
         "codex-otel",
         "codex-protocol",