[Fix] Fast child lifecycle relay uses configured API client

[daniel-lxs]

Summary

• configure the Fast child lifecycle relay client with the canonical MCP platform URL and run token
• preserve the existing auth-bypass header when the deployment requires it
• add regression coverage for the target URL and authentication headers

Why

The Roomote MCP subprocess receives ROOMOTE_PLATFORM_API_URL and ROOMOTE_CLOUD_TOKEN. The global worker SDK client instead reads TRPC_URL and AUTH_TOKEN, so child lifecycle relays fell back to localhost and failed before reaching the Fast parent.

This keeps the existing MCP environment contract and uses the same authenticated-header helper as other Roomote MCP platform calls.

Validation

• focused worker tests: 56 passed
• worker TypeScript check
• worker ESLint and formatting
• full pre-push suite: oxlint, residual lint, fast type checks, and knip
• end-to-end mock Slack smoke: Fast launched a fresh Local Docker task, then received and posted one child progress update and one child closeout without duplicates

[roomote-community]

No code issues found. See task

_{Reviewed cb5e055}