BYOK models should automatically appear in model selector without re-registration

### Feature summary

BYOK models should automatically appear in model selector without re-registration

### What problem are you trying to solve?

With the recent addition of Bring Your Own Key (BYOK) support in GitHub Copilot, models configured via BYOK should automatically be discoverable and available within the model selector experience.

Today, even after successfully configuring a BYOK provider and model within GitHub Enterprise AI Controls, those models do not automatically appear in the model picker within the Copilot application. Users must effectively re-register or manually configure models inside the app, which introduces unnecessary duplication and breaks the expected abstraction of provider-level configuration.

### Proposed solution

Models configured through BYOK (via provider configuration) within GitHub Enterprise should:

- Automatically populate in the Copilot model selector
- Be selectable without additional registration steps
- Stay in sync with provider configuration changes

### Workflow impact

- Introduces redundant configuration steps, requiring users to define models both at the provider level and again within the application
- Breaks seamless workflows between CLI, API, and UI where BYOK is otherwise already respected
- Slows down environment setup, especially in enterprise scenarios with multiple approved models
- Creates confusion for developers who expect configured models to be immediately usable across all Copilot experiences
- Increases operational overhead for platform teams managing standardized model configurations
- Disrupts air-gapped or controlled environments where re-registration may not be feasible or aligned with governance controls

### Installation context

_No response_

### Additional context

BYOK support already enables provider-level configuration where model identifiers are defined (e.g., via environment variables or provider endpoints). Extending this to the UI layer would align the experience across CLI, application, and enterprise configurations and reduce operational overhead.