### Check for existing issues

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

### What happened?

## Summary

When a team/org/user budget is exhausted in LiteLLM, the proxy returns HTTP 429 Budget Exceeded for model discovery endpoints such as GET /v1/models and GET /models.

This completely breaks OpenAI-compatible clients because they rely on these endpoints to populate available models before users can interact with the UI. As a result, all models become invisible - including free/self-hosted models that would not incur additional cost.

## What happened?

When a **team-level budget** (or organization/user budget) is fully exhausted, the LiteLLM proxy returns `HTTP 429 Budget Exceeded` on the `GET /v1/models` and `GET /models` endpoints. This completely blocks model discovery for any client using that team's API key.

This means **all models become invisible** to the client — including **free models** (zero-cost) that would not incur any additional spend. Clients like Open WebUI, Continue.dev, Cursor, Aider, TypingMind, BoltAI, LibreChat, and any other OpenAI-compatible client that calls `GET /v1/models` for model discovery will show **z…