### Check for existing issues

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

### What happened?

### What happened

With `router_settings.provider_budget_config` set, **every successful embedding request** through the Router logs a non-blocking

``` LiteLLM.LoggingError: [Non-Blocking] Exception occurred while success logging ... ValueError: custom_llm_provider is required ```

from `RouterBudgetLimiting.async_log_success_event` (`litellm/router_strategy/budget_limiter.py`). The embedding response itself is fine (HTTP 200). Chat completions are unaffected. The provider of the embedding deployment is irrelevant — it happens even for providers that have **no** budget configured.

Observed on `v1.82.0-stable`; the code paths below are unchanged on `main` as of 2026-06-12.

### Minimal reproduction

```python import asyncio

from litellm import Router

async def main() -> None: # Router must be built inside a running loop: with provider_budget_config # set, its __init__ schedules a periodic-sync task via asyncio.create_task. router = Router( model_list=[ { "model_name": "embed", …