[FEATURE] Push Claude Code subscription metrics to the gateway OTLP endpoint

[edenreich]

Summary

Follow-up to inference-gateway/inference-gateway#404 (client side), required by its acceptance criteria. The gateway now exposes an opt-in OTLP/HTTP metrics push endpoint (POST /v1/metrics, gateway PR inference-gateway/inference-gateway#409).

When infer runs with a Claude Code Max subscription, traffic goes infer CLI -> Claude Code -> Anthropic API and bypasses the gateway entirely, so no metrics are recorded. The CLI should parse Claude Code headless-mode logs and push usage/tool-call metrics to the gateway:

• Push OTLP JSON (or protobuf) to POST {gateway}/v1/metrics with Content-Type: application/json, delta temporality.
• Supported metrics: gen_ai.client.token.usage (delta sum per operation, with gen_ai.token.type = input|output), gen_ai.client.operation.duration, gen_ai.client.operation.time_to_first_chunk, gen_ai.execute_tool.duration (histograms, seconds; carry error.type on failures so tool success/failure is derivable), inference_gateway.tool_calls (delta monotonic sum).
• Attributes: gen_ai.provider.name, gen_ai.request.model, gen_ai.tool.name, gen_ai.tool.type, error.type, and set source to claude-code-subscription so subscription clients are labeled distinctly (anything else is dropped by the gateway's allowlist; source="gateway" is rewritten server-side).
• Forward the user's bearer token when the gateway runs with auth enabled; handle 403 (push disabled) gracefully.

Example payload: see "Pushing metrics (OTLP)" in the gateway's examples/docker-compose/monitoring/README.md.

Acceptance Criteria

• infer parses Claude Code headless-mode logs for token usage and tool-call outcomes
• Metrics are pushed to the gateway's POST /v1/metrics endpoint with source=claude-code-subscription
• Push failures (endpoint disabled, auth, network) degrade gracefully without breaking the session
• It's documented and tested