Feature Request: Status bar showing site/path/branch context while the AI is working (Full Agentic mode)

### What

Add a status bar in Studio, shown at least when using Full Agentic mode, that displays context about what the AI is currently doing: which site/repository it's pointed at, the file path it's reading or writing, and the current git branch.

Some ideas for what could live in the status bar:
- Current site / local WordPress install the AI is acting on
- File path currently being read/edited
- Current git branch
- The WP-CLI command or Studio action currently being executed

### Why

The new UI in Studio v2 feels aimed at newbies, mainly because when the chatbot is running there's no visible reference to which repository/site it's pointed at, or what it's actually doing to the filesystem/WordPress install.

As a developer, this matters because Studio is opened alongside VS Code, and Studio's Full Agentic mode is used to run actions against the local WordPress site while coding. Without a status bar, it's not possible to tell where the AI is writing, which makes it harder to trust and follow along with what's happening — the main appeal of Studio for a developer is precisely that the AI can interact with WordPress while you're coding, so visibility into "what is it touching right now" is important.

### How

No mockups yet — opening this issue mainly to capture the request. Happy to iterate on exactly which fields are most useful (site/path and git branch seem like the minimum) if that's helpful for scoping.

No files, tests, or entry points are named; start by locating the Studio v2 Full Agentic UI and the code that tracks the active site, filesystem activity, git branch, and WP-CLI or Studio actions. Done means agreeing on the minimum status-bar fields and displaying accurate current context while Full Agentic work is running.

Written by the indexing model from the issue text.