[FEATURE] enable vscode extention with 3.rd party servers

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Claude code (the CLI client), can work with these env variables and use corporate/custom API servers:

export ACME_LLM_API_KEY="XXXXXXXXXXXXXXXXXXXXXX"
export ANTHROPIC_API_KEY="dummy" 
export ANTHROPIC_BASE_URL="https://acme-llm-api-gateway.com/Anthropic"
export ANTHROPIC_CUSTOM_HEADERS="Ocp-Apim-Subscription-Key: ${ACME_LLM_API_KEY}"
# export ANTHROPIC_CUSTOM_HEADERS='{"Ocp-Apim-Subscription-Key":"'"${ACME_LLM_API_KEY}"'"}' # not this?
export ANTHROPIC_DEFAULT_SONNET_MODEL="claude-sonnet-4.5"
export ANTHROPIC_DEFAULT_OPUS_MODEL="claude-opus-4.1"
export ANTHROPIC_DEFAULT_HAIKU_MODEL="claude-3.5" # CC sends behind-the-scenes requests for things like, "make a title for this conversation." this makes these faster (optional)
export CLAUDE_CODE_DISABLE_NONESSENTIAL_TRAFFIC=1 # Disable telemetry and error reporting to anthropic

However, for the vscode extensions, even if I define all these envs, it still wants me to do the login to Anthropic's authentication system. I guess anthropic's server URLs are hard-coded into the vscode extension binary?
Some 3.rd party logins can be done (vertex ai, and some other), but it doesn't work with arbitrary server.

I can do this trick: first login to anthropic account and then enable these env variables .. this makes the vscode extension confused and the request hang.

Proposed Solution

Please enable the vscode extension to use arbitrary 3.rd party servers! :D