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/lmstudio/Cargo.toml

@@ -12,8 +12,8 @@ doctest = false
 
 [dependencies]
 codex-core = { path = "../core" }
+codex-http-client = { workspace = true }
 codex-model-provider-info = { path = "../model-provider-info" }
-reqwest = { version = "0.12", features = ["json", "stream"] }
 serde_json = "1"
 tokio = { version = "1", features = ["rt"] }
 tracing = { version = "0.1.44", features = ["log"] }