[codex] Use background task auth for additional backend calls (#18260)

## Summary

Splits the larger PR4.1 background task auth rollout by moving
additional backend/control-plane call sites into this downstream PR.

This PR keeps callers on the same design as PR4.1: most code asks
`AuthManager` for the default ChatGPT backend authorization header, and
`AuthManager` decides bearer vs background AgentAssertion internally.
Task-pinned inference auth remains separate because it needs the
thread's registered task id.

## Stack

- PR1: https://github.com/openai/codex/pull/17385 - add
`features.use_agent_identity`
- PR2: https://github.com/openai/codex/pull/17386 - register agent
identities when enabled
- PR3: https://github.com/openai/codex/pull/17387 - register agent tasks
when enabled
- PR3.1: https://github.com/openai/codex/pull/17978 - persist and
prewarm registered tasks per thread
- PR4: https://github.com/openai/codex/pull/17980 - use task-scoped
`AgentAssertion` for downstream calls
- PR4.1: https://github.com/openai/codex/pull/18094 - introduce
AuthManager-owned background/control-plane `AgentAssertion` auth
- PR4.2: this PR - use background task auth for additional
backend/control-plane calls

## What Changed

- pass full authorization header values through backend-client and
cloud-tasks-client call paths where needed
- move ChatGPT client, cloud requirements, cloud tasks, thread-manager,
and models-manager background auth usage into this downstream slice
- make app-server remote control enrollment/websocket auth ask
`AuthManager` for the local backend authorization header instead of
threading a background auth mode through transport options
- keep the same feature-gated bearer fallback behavior from PR4.1

## Validation

- `just fmt`
- `cargo check -p codex-core -p codex-login -p codex-analytics -p
codex-app-server -p codex-cloud-requirements -p codex-cloud-tasks -p
codex-models-manager -p codex-chatgpt -p codex-model-provider -p
codex-mcp -p codex-core-skills`
- `cargo test -p codex-login agent_identity`
- `cargo test -p codex-model-provider bearer_auth_provider`
- `cargo test -p codex-core agent_assertion`
- `cargo test -p codex-app-server remote_control`
- `cargo test -p codex-cloud-requirements fetch_cloud_requirements`
- `cargo test -p codex-models-manager manager::tests`
- `cargo test -p codex-chatgpt`
- `cargo test -p codex-cloud-tasks`
- `just fix -p codex-core -p codex-login -p codex-analytics -p
codex-app-server -p codex-cloud-requirements -p codex-cloud-tasks -p
codex-models-manager -p codex-chatgpt -p codex-model-provider -p
codex-mcp -p codex-core-skills`
- `just fix -p codex-app-server`
- `git diff --check`

codex-rs/app-server/src/codex_message_processor.rs

@@ -2013,12 +2013,28 @@ impl CodexMessageProcessor {
             });
         }
 
-        let client = BackendClient::from_auth(self.config.chatgpt_base_url.clone(), &auth)
+        let authorization_header_value = self
+            .auth_manager
+            .chatgpt_authorization_header_for_auth(&auth)
+            .await;
+        let mut client = BackendClient::new(self.config.chatgpt_base_url.clone())
+            .map(|client| {
+                client.with_user_agent(codex_login::default_client::get_codex_user_agent())
+            })
             .map_err(|err| JSONRPCErrorError {
                 code: INTERNAL_ERROR_CODE,
                 message: format!("failed to construct backend client: {err}"),
                 data: None,
             })?;
+        if let Some(authorization_header_value) = authorization_header_value {
+            client = client.with_authorization_header_value(authorization_header_value);
+        }
+        if let Some(account_id) = auth.get_account_id() {
+            client = client.with_chatgpt_account_id(account_id);
+        }
+        if auth.is_fedramp_account() {
+            client = client.with_fedramp_routing_header();
+        }
 
         let snapshots = client
             .get_rate_limits_many()