Support model discovery and in-session switching for BYOM providers

Describe the feature or problem you'd like to solve

Copilot CLI's BYOM configuration currently requires a single COPILOT_MODEL value. When using Google Vertex AI through its OpenAI-compatible endpoint and gcloud authentication, users must restart Copilot CLI to change models.

This is a gap in the model-selection experience: a Copilot subscription exposes multiple GitHub-hosted models through the model picker, but BYOM is effectively one provider/model per process. A subscription-based model experience should support many available models behind one provider configuration (many-to-one), rather than requiring a separate CLI process for each model (one-to-one).

Think BYOMS (Bring your own model subscription) instead of a single BYOM.

Proposed solution

Support model discovery and model selection for BYOM providers, including Vertex AI:

• Discover models exposed by the configured provider, or allow a provider-specific model catalog/configuration.
• Show available BYOM models in the same model picker used for GitHub-hosted models.
• Allow switching models during a Copilot CLI session without restarting the CLI.
• Preserve provider-level authentication and endpoint settings while changing only the selected model.
• Support short-lived Google OAuth access tokens from gcloud/ADC without requiring users to manually restart the CLI when a token refresh is needed.

For Vertex AI, this should work with the OpenAI-compatible endpoint and Google Cloud authentication, while respecting the models enabled and authorized for the user's project and region.

Example prompts or workflows

copilot
# Model picker shows GitHub models plus authorized Vertex AI models.
# Selecting a different model changes the active BYOM model in the current session.

Additional context

Current workaround:

$env:COPILOT_PROVIDER_BASE_URL = "https://LOCATION-aiplatform.googleapis.com/v1/projects/PROJECT/locations/LOCATION/endpoints/openapi"
$env:COPILOT_PROVIDER_API_KEY = gcloud auth print-access-token
$env:COPILOT_MODEL = "google/gemini-2.5-flash"
copilot

Changing COPILOT_MODEL requires restarting Copilot CLI. Related issue: #3399 (custom headers for BYOK).

The issue names the COPILOT_MODEL, COPILOT_PROVIDER_BASE_URL, and COPILOT_PROVIDER_API_KEY configuration entry points and the copilot command; start by tracing how those values feed the current model picker and session. Done means the picker can show authorized Vertex AI/BYOM models, switch them without restarting, preserve endpoint and authentication settings, and refresh short-lived gcloud/ADC tokens.