### Check for existing issues

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

### What happened?

## Summary

When using the `websearch_interception` callback with models behind a custom Anthropic-compatible endpoint (e.g. `api_base: https://my-proxy.example.com/anthropic`), the follow-up request in the agentic loop **ignores the deployment's `api_base` and `api_key`** and falls back to `https://api.anthropic.com` with `ANTHROPIC_API_KEY` env var. This results in a `401 Unauthorized` error, which then triggers a **deployment cooldown cascade** that blocks all subsequent requests to that model for the cooldown duration (default 5s).

## Environment

- LiteLLM Proxy (latest `main` branch) - Model configured with `custom_llm_provider: anthropic` and a custom `api_base` (e.g. Tencent Cloud Anthropic-compatible endpoint) - `websearch_interception` callback enabled in `litellm_settings.callbacks` - Request via Anthropic Messages API: `POST /v1/messages?beta=true`

## Steps to Reproduce

1. Add a model deployment with `custom_llm_provider: anthropic` and a custom `api_base` (not `api.anthropic.com`), e.g.:

```yaml model_list: - model_…