Trace the local Claude Code TS request path and align the Rust client with its
non-OAuth direct-request behavior. The Rust client now resolves the message base
URL from ANTHROPIC_BASE_URL, uses ANTHROPIC_API_KEY for x-api-key, and sends
ANTHROPIC_AUTH_TOKEN as a Bearer Authorization header when present.
Constraint: Must match the local Claude Code source request/auth split, not inferred behavior
Rejected: Treat ANTHROPIC_AUTH_TOKEN as the x-api-key source | diverges from local TS client path
Confidence: high
Scope-risk: narrow
Reversibility: clean
Directive: Keep direct /v1/messages auth handling aligned with src/services/api/client.ts and src/utils/auth.ts when changing env precedence
Tested: cargo test -p api; cargo run -p rusty-claude-cli -- prompt "say hello"
Not-tested: Non-default proxy transport features beyond ANTHROPIC_BASE_URL override
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