[BUG] [Claude Code VS Code Extension]Keep asking for login even if I am using third party provider [vertex]

Preflight Checklist

• [x] I have searched existing issues and this hasn't been reported yet
• [x] This is a single bug report (please file separate reports for different bugs)
• [x] I am using the latest version of Claude Code

What's Wrong?

In the past, I am using an older version of "Claude Code for VS code" extension together with "Claude Code" which worked perfect. I used vertex as the model provider for Claude Code.

One week ago I upgraded the extension to the latest version (2.0.10) and followed the instruction to update the VS Code setting as below

{
    "claude-code.environmentVariables": [      
      {
          "ANTHROPIC_MODEL": "claude-sonnet-4-5@20250929",
          "CLAUDE_CODE_USE_VERTEX": "1",
          "CLOUD_ML_REGION": "us-east5",
          "ANTHROPIC_VERTEX_PROJECT_ID": "XXX"
      }
    ]
}

I am sure this setting works because claude code still worked. However, ""Claude Code for VS code" extension" keep asking for login. Please kindly take a look and fix it ASAP.

What Should Happen?

Given the instruction , there is no need for login if I am using third-party provider.

Error Messages/Logs

No error message.

Steps to Reproduce

1. Upgraded the extension to the latest version (2.0.10)
2. Followed the instruction to update the VS Code setting as below

{
    "claude-code.environmentVariables": [      
      {
          "ANTHROPIC_MODEL": "claude-sonnet-4-5@20250929",
          "CLAUDE_CODE_USE_VERTEX": "1",
          "CLOUD_ML_REGION": "us-east5",
          "ANTHROPIC_VERTEX_PROJECT_ID": "XXX"
      }
    ]
}

1. completely quit vscode and re-enter.

Claude Model

None

Is this a regression?

Yes, this worked in a previous version

Last Working Version

_No response_

Claude Code Version

2.0.1

Platform

Google Vertex AI

Operating System

macOS

Terminal/Shell

iTerm2

Additional Information

_No response_