Make direct_access API endpoint resilient to upstream LLM provider outages
## Problem
An outage at Anthropic/Claude caused gitlab monolith's call to the following endpoints to halt for 30 seconds and causing puma worker to saturate.
- POST `/v1/code/user_access_token`
This was requested from Rails:
- POST `/api/v4/code_suggestions/direct_access`
- POST `/api/v4/ai/third_party_agents/direct_access`
See https://log.gprd.gitlab.net/app/r/s/tj0pq for example of those slow requests
## Root Cause
The direct_access API endpoints lacks resilience mechanisms to handle upstream provider outages. When the Claude API experiences issues, requests accumulate and exhaust available worker threads.
## Solution
Improve the resilience of the direct_access API endpoint to better handle upstream LLM provider outages and prevent worker saturation.
## Related Issues
- [#21728 Incident Review: Puma Saturation for ai-assisted](https://gitlab.com/gitlab-com/gl-infra/production/-/work_items/21728)
- [#28660 Handle cascading error from the claude API better](https://gitlab.com/gitlab-com/gl-infra/production-engineering/-/work_items/28660)
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