Enable Proxying Copilot Requests to External LLMs (Claude, etc.)

**Is your feature request related to a problem? Please describe.**
I want to test the GitHub Copilot extension with APls I already have. While there are self-hosted LLM replacements for Copilot, they don't seem to support proxying to remote inference endpoints.

**Describe the solution you'd like**
I want to use this project to create a Copilot inference tool. This tool should connect to a remote LLM API (like Claude) using my authentication. Then, I'd like to use GitHub Copilot as normal, but with requests routed to the remote LLM instead of a local one

**Describe alternatives you've considered**
A clear and concise description of any alternative solutions or features you've considered.

**Additional context**
Add any other context or screenshots about the feature request here.