Support VS Code's browser-sharing API so Claude can validate web UI changes

Summary

VS Code recently added a browser-sharing capability (release notes: "Sharing browser tabs with agents") that lets the built-in Copilot agent read and interact with tabs the user explicitly shares — DOM, screenshots, console output, navigation. This is the right primitive for an agent doing web/UI work: edit code, reload, verify, iterate in one turn.

Claude Code's VS Code integration currently only exposes mcp__ide__executeCode and mcp__ide__getDiagnostics. There's no way for Claude to see a running app the user has open, even when the user wants to share it. Today the workarounds are screenshots, paste, WebFetch (no auth, no JS), or driving headless Playwright via Bash — all clunkier than what Copilot agent mode now gets for free.

Ask

Extend the Claude Code VS Code extension to consume VS Code's browser-sharing API so Claude can request shares and read shared tabs the same way. User stays in control of what's shared (matches the existing Copilot UX).

Why it matters

Unblocks real-time web/UI validation — frontend bugs, design iteration, layout/styling refinement — without round-tripping through screenshots.