### Check for existing issues

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

### What happened?

A bug happenedAfter updating a DB-backed `auto_router/complexity_router` model through `/model/{model_id}/update`, the model still exists in `LiteLLM_ProxyModelTable`, but it no longer appears in `/v2/model/info` or `/v2/model/info?include_team_models=true&page=1&size=50`.

This only happens after updating the router. Before the update, the router is returned by the model info endpoint.

### Steps to Reproduce

1. Create a DB-backed auto router model with: ```json { "model_name": "auto_router", "litellm_params": { "model": "auto_router/complexity_router", "complexity_router_config": { "tiers": { "SIMPLE": "qwen3.7-plus", "MEDIUM": "qwen3.7-plus", "COMPLEX": "qwen3.7-max", "REASONING": "qwen3.7-max" } }, "complexity_router_default_model": "qwen3.7-plus", "rpm": 9, "tpm": 10 }, "model_info": { "id": "51ea2cad-a498-49bf-83b4-1bd397d509b5", "key": "auto_router/complexity_router", "litellm_provider": "auto_router", "db_model": true } } ``` 2. Confirm the m…