[BUG] Claude Code cannot login if connecting to claude.ai shows Cloudflare verification

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?

When connecting claude.ai prompting Cloudflare verification, the Claude Code CLI cannot login at all even though I can use it normally in web client.
Please fix this :)

What Should Happen?

The Claude Code CLI should also pass the auth process and can be used properly.

Error Messages/Logs

Steps to Reproduce

1. Open Cluade Code.
2. It shows error message.

Claude Model

None

Is this a regression?

Yes, this worked in a previous version

Last Working Version

_No response_

Claude Code Version

2.0.22

Platform

Anthropic API

Operating System

macOS

Terminal/Shell

Terminal.app (macOS)

Additional Information

<img width="544" height="341" alt="Image" src="https://github.com/user-attachments/assets/1dd76819-da05-4b08-9b0b-b97a87c4e3e6" />

<img width="634" height="378" alt="Image" src="https://github.com/user-attachments/assets/da5f726f-b134-4c55-95e1-90fb658e07f0" />