[FEATURE] "Update branch" action in the desktop app's CI monitoring popover

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

In the Claude Code desktop app, each session's PR bar has a CI monitoring popover (In progress / Passed / Skipped counts, plus the Auto-fix, Auto-merge when ready, and Auto-archive toggles). When several PRs are open at once and merge one after another, the later ones fall behind `main` and GitHub asks for the branch to be updated before merging. Today the popover shows that a check is pending or failing but offers no way to bring the branch up to date; I have to leave the app, open the PR on github.com, click "Update branch", and come back — or ask Claude in chat to merge `main` in and push.

This happens on every multi-PR session, which is exactly the workflow the PR bar is designed for.

### Proposed Solution

Add an "Update branch" action to the CI monitoring popover (next to the Auto-fix / Auto-merge / Auto-archive controls). It would merge the base branch into the PR's head branch and push — the same operation as GitHub's own "Update branch" button — using a merge commit, never a rebase or force-push, and only when GitHub reports the branch as behind. Optionally, a companion toggle "Keep branch up to date" could do this automatically whenever the base moves, so queued PRs stay mergeable without manual clicks.

### Alternative Solutions

Workarounds today: GitHub's "Update branch" button on the PR page, or asking Claude in the session to `git merge origin/main` and push. Both work but break the flow the popover otherwise handles in place. Enabling a GitHub merge queue avoids the problem but requires branch protection rules many small repos don't have.

### Priority

Medium - Would be very helpful

### Feature Category

Other

### Use Case Example

1. Open four small PRs from one session (they're independent but share a base).
2. Merge #1 on GitHub; #2–#4 are now behind main.
3. In the app, the CI popover for #2 shows the state, but the only way to update the branch is to leave for github.com or ask Claude in chat.
Expected: click "Update branch" in the popover (or have it happen automatically), then merge #2, and so on.

### Additional Context

_No response_

Start by locating the desktop app's CI monitoring popover described in the issue and trace how it reads PR status and exposes the existing Auto-fix, Auto-merge, and Auto-archive controls. Determine how the app could invoke GitHub's Update branch operation only when the branch is behind, then cover the click behavior, merge-commit and push constraints, and the optional automatic mode with tests.

Written by the indexing model from the issue text.