feat(aio): add Claude Agent SDK integration to the JS SDK

Context

The Claude Agent SDK integration for LLM Analytics only exists in the Python SDK (posthog.ai.claude_agent_sdk, added in #53010). TypeScript users building on @anthropic-ai/claude-agent-sdk have no way to get traces into LLM Analytics today.

@posthog/ai currently covers Anthropic, OpenAI, OpenAI Agents, Gemini, LangChain, Vercel AI SDK and Google ADK. The existing Anthropic wrapper does not help here: the Agent SDK runs Claude Code itself, so requests never pass through the Anthropic client.

This has been asked for by a customer.

What

A claude-agent-sdk integration in posthog-js' packages/ai, mirroring the Python one:

• a query() drop-in replacement for the SDK's query()
• a stateful client equivalent for multi-turn conversations
• $ai_generation per LLM turn (model, tokens, cache metrics, cost, latency, input/output)
• $ai_span per tool use
• $ai_trace per query, with aggregate cost and latency

Why it should be small

The Python integration does not patch HTTP. It consumes the Agent SDK's own message stream: message_start to message_stop stream events delimit one generation, tool blocks become spans, and the result message closes the trace. The TypeScript SDK exposes the same surface, with includePartialMessages: true for the per-turn stream events and total_cost_usd plus usage on the result message. So the port is mostly a translation of posthog/ai/claude_agent_sdk/processor.py.

Reference: PostHog/posthog-python#477.

Notes

• Wrap all instrumentation in try/catch, so PostHog failures never interrupt the underlying query.
• Docs page ai-observability/installation/claude-agent-sdk is Python-only and needs a TypeScript tab.

Start in posthog-js/packages/ai and read posthog/ai/claude_agent_sdk/processor.py alongside the TypeScript Agent SDK stream surface, including includePartialMessages and the result message. Add the query wrapper, stateful client, and event instrumentation described in the issue, then update ai-observability/installation/claude-agent-sdk with a TypeScript tab. Done means generations, tool spans, query traces, metrics, and costs are captured without interrupting the underlying query.