[FEATURE] Google Vertex support for Remote Control features

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

We use Google Vertex as our billing model for Claude Code usage, it would be great if we can support Remote Control via that account type rather than having a claude.ai subscription.

Proposed Solution

Google Vertex should be seen as a logged an path and continue with Remote Control as intended

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

CLI commands and flags

Use Case Example

_No response_

Additional Context

_No response_