[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/cloud-tasks-client/src/http.rs

@@ -37,6 +37,11 @@ impl HttpClient {
         self
     }
 
+    pub fn with_authorization_header_value(mut self, value: impl Into<String>) -> Self {
+        self.backend = self.backend.clone().with_authorization_header_value(value);
+        self
+    }
+
     pub fn with_user_agent(mut self, ua: impl Into<String>) -> Self {
         self.backend = self.backend.clone().with_user_agent(ua);
         self
@@ -47,6 +52,11 @@ impl HttpClient {
         self
     }
 
+    pub fn with_fedramp_routing_header(mut self) -> Self {
+        self.backend = self.backend.clone().with_fedramp_routing_header();
+        self
+    }
+
     fn tasks_api(&self) -> api::Tasks<'_> {
         api::Tasks::new(self)
     }