[FEATURE] Support Channels with Vertex AI / third-party auth providers

Summary

Channels currently require claude.ai login. Users on Vertex AI (or Bedrock/API key) auth cannot use channels at all.

Use case

I use Claude Code via Vertex AI (authMethod: "third_party", apiProvider: "vertex") and would like to use channels (Telegram, Discord, custom webhooks) for remote interaction with my sessions.

Current behavior

Channels require authMethod: "claude_ai". No error or workaround is provided for third-party auth users.

Requested behavior

Support channels with Vertex AI, Bedrock, and API key authentication methods.