Support alternative websocket API (#10861)

**Test plan**

```
cargo build -p codex-cli && RUST_LOG='codex_api::endpoint::responses_websocket=trace,codex_core::client=debug,codex_core::codex=debug' \
  ./target/debug/codex \
    --enable responses_websockets_v2 \
    --profile byok \
    --full-auto
```

codex-rs/core/src/config/mod.rs

@@ -2685,25 +2685,27 @@ profile = "project"
     }
 
     #[test]
-    fn responses_websockets_feature_does_not_change_wire_api() -> std::io::Result<()> {
-        let codex_home = TempDir::new()?;
-        let mut entries = BTreeMap::new();
-        entries.insert("responses_websockets".to_string(), true);
-        let cfg = ConfigToml {
-            features: Some(crate::features::FeaturesToml { entries }),
-            ..Default::default()
-        };
+    fn responses_websocket_features_do_not_change_wire_api() -> std::io::Result<()> {
+        for feature_key in ["responses_websockets", "responses_websockets_v2"] {
+            let codex_home = TempDir::new()?;
+            let mut entries = BTreeMap::new();
+            entries.insert(feature_key.to_string(), true);
+            let cfg = ConfigToml {
+                features: Some(crate::features::FeaturesToml { entries }),
+                ..Default::default()
+            };
 
-        let config = Config::load_from_base_config_with_overrides(
-            cfg,
-            ConfigOverrides::default(),
-            codex_home.path().to_path_buf(),
-        )?;
+            let config = Config::load_from_base_config_with_overrides(
+                cfg,
+                ConfigOverrides::default(),
+                codex_home.path().to_path_buf(),
+            )?;
 
-        assert_eq!(
-            config.model_provider.wire_api,
-            crate::model_provider_info::WireApi::Responses
-        );
+            assert_eq!(
+                config.model_provider.wire_api,
+                crate::model_provider_info::WireApi::Responses
+            );
+        }
 
         Ok(())
     }