Remote Control should support AWS Bedrock authentication

Feature Request

Remote Control currently requires a claude.ai subscription and does not work with AWS Bedrock or API key authentication. Running claude remote-control with Bedrock credentials
returns:

``
Error: You must be logged in to use Remote Control.
Remote Control is only available with claude.ai subscriptions.
``

## Use Case

We run Claude Code in sandboxed EC2 instances (CDEs) that use Bedrock via VPC endpoint for all model access. Being able to monitor and interact with these sessions from a phone via
Remote Control would be very useful, but the claude.ai login requirement blocks this.

Enterprise/security-conscious deployments are likely using Bedrock or API keys rather than claude.ai subscriptions, so this restriction effectively locks out the users who would
benefit most from remote session management.

## Request

Support Bedrock (and API key) authentication for Remote Control sessions.