[codex] Use standalone tools for Responses Lite (#26490)

## Summary

Responses Lite does not execute hosted Responses tools, so models using
it must route web search and image generation through Codex-owned
executors & standalone Response's API endpoints.

This PR is stacked on #26487.

## Validation

- `cargo test -p codex-core responses_lite_ --lib`
- `cargo test -p codex-core
standalone_executors_remain_hidden_without_flags_or_responses_lite
--lib`
- `cargo test -p codex-core
hosted_tools_follow_provider_auth_model_and_config_gates --lib`
- `cargo test -p codex-web-search-extension -p
codex-image-generation-extension`
- `cargo test -p codex-app-server --test all standalone_`
- `cargo fmt --all -- --check`

codex-rs/core/tests/common/test_codex.rs

@@ -23,12 +23,14 @@ use codex_core::thread_store_from_config;
 use codex_exec_server::CreateDirectoryOptions;
 use codex_exec_server::ExecutorFileSystem;
 use codex_exec_server::RemoveOptions;
+use codex_extension_api::ExtensionRegistry;
 use codex_extension_api::empty_extension_registry;
 use codex_login::CodexAuth;
 use codex_model_provider_info::ModelProviderInfo;
 use codex_model_provider_info::built_in_model_providers;
 use codex_models_manager::bundled_models_response;
 use codex_protocol::models::PermissionProfile;
+use codex_protocol::openai_models::ModelInfo;
 use codex_protocol::openai_models::ModelsResponse;
 use codex_protocol::protocol::AskForApproval;
 use codex_protocol::protocol::EventMsg;
@@ -216,6 +218,7 @@ pub struct TestCodexBuilder {
     cloud_config_bundle: Option<CloudConfigBundleLoader>,
     user_shell_override: Option<Shell>,
     exec_server_url: Option<String>,
+    extensions: Arc<ExtensionRegistry<Config>>,
 }
 
 impl TestCodexBuilder {
@@ -239,6 +242,26 @@ impl TestCodexBuilder {
         })
     }
 
+    pub fn with_model_info_override<T>(self, model: &str, override_model_info: T) -> Self
+    where
+        T: FnOnce(&mut ModelInfo) + Send + 'static,
+    {
+        let model = model.to_string();
+        self.with_config(move |config| {
+            let model_catalog = config.model_catalog.get_or_insert_with(|| {
+                bundled_models_response()
+                    .unwrap_or_else(|err| panic!("bundled models.json should parse: {err}"))
+            });
+            let model_info = model_catalog
+                .models
+                .iter_mut()
+                .find(|model_info| model_info.slug == model)
+                .unwrap_or_else(|| panic!("{model} should exist in the configured model catalog"));
+            override_model_info(model_info);
+            config.model = Some(model);
+        })
+    }
+
     pub fn with_pre_build_hook<F>(mut self, hook: F) -> Self
     where
         F: FnOnce(&Path) + Send + 'static,
@@ -280,6 +303,11 @@ impl TestCodexBuilder {
         self
     }
 
+    pub fn with_extensions(mut self, extensions: Arc<ExtensionRegistry<Config>>) -> Self {
+        self.extensions = extensions;
+        self
+    }
+
     pub fn with_windows_cmd_shell(self) -> Self {
         if cfg!(windows) {
             self.with_user_shell(get_shell_by_model_provided_path(&PathBuf::from("cmd.exe")))
@@ -473,7 +501,7 @@ impl TestCodexBuilder {
             codex_core::test_support::auth_manager_from_auth(auth.clone()),
             SessionSource::Exec,
             Arc::clone(&environment_manager),
-            empty_extension_registry(),
+            Arc::clone(&self.extensions),
             /*analytics_events_client*/ None,
             thread_store,
             state_db.clone(),
@@ -1041,6 +1069,7 @@ pub fn test_codex() -> TestCodexBuilder {
         cloud_config_bundle: None,
         user_shell_override: None,
         exec_server_url: None,
+        extensions: empty_extension_registry(),
     }
 }