[BUG] False auth conflict warning when using third-party ANTHROPIC_BASE_URL with both API key and /login managed key

Environment

• Platform: Windows 11
• Claude Code: latest

Scenario

Two auth methods are set at the same time, targeting different servers:

1. ANTHROPIC_AUTH_TOKEN — a DeepSeek API key, used with ANTHROPIC_BASE_URL=https://api.deepseek.com/anthropic. All model calls go to DeepSeek.
2. /login managed key — an Anthropic OAuth token, used for claude.ai platform features (DesignSync, etc.)

Problem

Claude Code shows this warning on startup:

Both ANTHROPIC_AUTH_TOKEN and /login managed key set · auth may not work as expected

Why this is a false positive

• ANTHROPIC_AUTH_TOKEN targets api.deepseek.com (third-party)
• /login managed key targets api.anthropic.com (Anthropic platform)
• The two keys go to different servers and do not actually conflict. Everything works fine in practice.

Suggested fix

When ANTHROPIC_BASE_URL is set to a non-Anthropic endpoint, ANTHROPIC_AUTH_TOKEN and the /login managed key should be allowed to coexist. They serve different purposes:

| Auth method | Target server | Purpose |
|---|---|---|
| ANTHROPIC_AUTH_TOKEN | Third-party (e.g. api.deepseek.com) | Model calls |
| /login managed key | api.anthropic.com | Platform features (DesignSync, etc.) |

Alternatively, provide a configuration option to explicitly set priority or suppress this warning.

Actual impact

Functionally everything works, but the warning appears on every startup, which degrades the user experience.