Add AI Badgr as an OpenAI-compatible backend example

[michaelmanly]

Add documentation for using AI Badgr as an OpenAI-compatible backend

This PR adds a small documentation section showing how to point existing OpenAI client code at AI Badgr, which follows the same API surface (chat completions, streaming, JSON mode).

AI Badgr uses optimized open-source models to provide more affordable inference and also supports optional private/self-hosted modes for teams that need additional data control.

What's Included

• A new README section showing how to override base_url and api_key
• Language-specific examples:
  • JavaScript/TypeScript and cURL (for JS repos)
• Notes explaining that AI Badgr works with standard OpenAI SDKs without code changes

Why This May Be Useful

Some users of this project work with OpenAI-compatible backends (self-hosted or third-party).

This example gives them a straightforward way to use AI Badgr as another compatible option while keeping their existing client code and request format.

Technical Notes

AI Badgr supports:

• /v1/chat/completions with streaming
• JSON mode (response_format)
• Standard OpenAI SDKs (openai clients for Python and JS)
• The same request/response schema used by OpenAI

This PR only adds optional documentation and does not modify any existing functionality.

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