[FEATURE] Add billing info (API usage / subscription tier) to statusbar

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

Description

Add support for displaying billing and subscription information in the statusbar, similar to the existing rate limit display.

Motivation

Claude Code already knows billing and presents it when launched, e.g. API billing, Claude Max.
Add that to statusline, which might be helpful for long threads.

Proposed Solution

Add new billing object with subscription_tier field

{
  "billing": {
    "subscription_tier": "pro"
  }
}

Alternative Solutions

_No response_

Priority

Low - Nice to have

Feature Category

Interactive mode (TUI)

Use Case Example

_No response_

Additional Context

_No response_