[BUG] CLI API Key Helper execution environment does not update ANTHROPIC_BASE_URL specified in settings.json

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 Claude Code CLI invokes an apiKeyHelper specified in a settings.json file and an ANTHROPIC_BASE_URL is specified in the settings.json file, the key helper execution environment does not include the ANTHROPIC_BASE_URL value from the settings.json file.

What Should Happen?

Claude Code should include the ANTHROPIC_BASE_URL variable in the environment of the specified key helper.

Error Messages/Logs

Steps to Reproduce

Claude CLI will not work properly with these settings, but will generate an output file that shows the problematic behavior.

1. Prepare your environment with any ANTHROPIC_BASE_URL:

````
export ANTHROPIC_BASE_URL=BASEENV

````

1. Create a .claude/settings.json file with a key helper and :

{
    "apiKeyHelper": "printenv | grep ANTHROPIC > helper.env",
    "env": {
        "ANTHROPIC_BASE_URL": "JSON"
    }
}
````
3. Invoke Claude Code CLI. The output helper.env file will show "BASEENV" instead of the desired "JSON"

### Claude Model

Not sure / Multiple models

### Is this a regression?

I don't know

### Last Working Version

_No response_

### Claude Code Version

2.1.47

### Platform

AWS Bedrock

### Operating System

macOS

### Terminal/Shell

Terminal.app (macOS)

### Additional Information

Other env variables in settings.json seem to override just fine. And this works as expected in the VSCode plugin.