### Check for existing issues

- [x] I have searched the existing issues and checked that my issue is not a duplicate.

### What happened?

OpenRouter models configured with gateway-wide web search via `litellm_params.extra_body.tools` (`openrouter:web_search`, optionally `openrouter:datetime`) are marked **unhealthy** by LiteLLM background health checks, while normal chat completions on the same deployment succeed.

**Expected:** Health checks pass for a deployment that can complete real chat requests, or probes should not inherit server-tool `extra_body` that changes request semantics.

**Actual:** Removing `extra_body` makes health checks pass. Adding only `openrouter:web_search` in `extra_body` makes health checks fail again. Increasing `model_info.health_check_max_tokens` did not help.

**Environment**

- LiteLLM v1.89.2 (production) and v1.89.3 (local docker) — same behavior - Provider: OpenRouter - Example models: `openrouter/anthropic/claude-haiku-4.5`, `openrouter/google/gemini-3-flash-preview` - Models stored in DB via Admin UI (`STORE_MODEL_IN_DB=True`)

**Suspected cause**

`litellm/proxy/health_check.py` function `_update_litellm_params_for_health_check()` overrides `m…