"Update available" notification persists on latest Homebrew version (v2.1.109) — regression from #43251 fix

Description

The "Update available! Run: brew upgrade claude-code" notification appears in the status line, but brew upgrade has no newer version available. The notification directs users to a command that can't resolve it.

It appears the update check is comparing against a version (2.1.116) that exists on npm but hasn't landed in the Homebrew cask yet. This is the same root cause as #43251, which was reportedly fixed in v2.1.101 to compare against the actual cask version on formulae.brew.sh.

Steps to reproduce

1. brew upgrade claude-code → confirms already on latest Homebrew version (2.1.109)
2. Launch claude
3. "Update available! Run: brew upgrade claude-code" shown in lower-right status line
4. Running brew upgrade again does nothing — 2.1.109 is the latest available via Homebrew

Environment

$ claude --version
2.1.109 (Claude Code)

$ brew info claude-code
==> claude-code (Claude Code): 2.1.109
Installed
/opt/homebrew/Caskroom/claude-code/2.1.109 (201.7MB)

• Latest available (npm): 2.1.116
• Latest available (Homebrew): 2.1.109
• Platform: macOS Darwin 25.5.0 (Apple Silicon)
• Install method: Homebrew

Expected

For Homebrew installs, the update check should either:

1. Compare against the latest Homebrew cask version (not npm), or
2. Suggest npm install -g @anthropic-ai/claude-code@latest if the Homebrew version lags behind