feat: enable Bedrock fallback via LLM gateway

[joshsny]

Summary

• Adds x-posthog-use-bedrock-fallback: true to requests sent through the LLM gateway so AWS Bedrock is used as a fallback whenever Anthropic returns a 5xx.
• Header is injected in two places:
  • AuthProxyService — covers every request the Claude Agent SDK subprocess makes (its LLM_GATEWAY_URL points at this local proxy).
  • LlmGatewayService — direct gateway calls (e.g. title generation).
• Mirrors the approach in posthog/wizard, which sets the same header via ANTHROPIC_CUSTOM_HEADERS. Centralising injection in the proxy keeps it independent of Claude Agent SDK env-var support and applies uniformly to all forwarded traffic.

Test plan

• pnpm --filter code typecheck passes
• pnpm --filter code test — 1164 tests pass
• Smoke test live: trigger an Anthropic 5xx and confirm Bedrock fallback path engages on the gateway side

---

Created with PostHog Code

[greptile-apps]

Prompt To Fix All With AI

Fix the following 2 code review issues. Work through them one at a time, proposing concise fixes.

---

### Issue 1 of 2
packages/agent/src/adapters/claude/session/options.ts:104-109
**Possible duplicate header injection**

If the operator's environment already contains `x-posthog-use-bedrock-fallback: true` in `ANTHROPIC_CUSTOM_HEADERS` (e.g. they've adopted the `posthog/wizard` approach at the process level), the header will appear twice in the child's env: once from the preserved existing value and once from the new append. Most gateways silently ignore duplicates, but the behaviour is technically undefined. A simple guard before appending would prevent this.

### Issue 2 of 2
packages/agent/src/adapters/claude/session/options.ts:104-120
**No tests for the new header-merging behaviour**

`options.test.ts` exercises agent registration and override but has nothing covering the `ANTHROPIC_CUSTOM_HEADERS` composition logic — neither the base case (env var absent → header set alone) nor the merge case (env var present → header appended). Given that the merge logic involves string manipulation that could silently produce wrong output (e.g. wrong separator, duplicate header), a pair of unit tests here would be cheap insurance and aligns with the team's preference for parameterised tests.

_{Reviews (1): Last reviewed commit: "refactor: move Bedrock fallback header t..." | Re-trigger Greptile}