The Anthropic SDK's messages.stream() is a synchronous context manager,
not async. For async streaming, we need to use messages.create(..., stream=True)
which returns an async iterator of ServerSentEvent objects.

Changed from:
    async with client.messages.stream(**request_kwargs) as stream:
To:
    stream = await client.messages.create(**request_kwargs, stream=True)
    async for event in stream: