[FEATURE] Add "Copy" button under assistant responses (like Zed's AI panel)

### 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

Currently there's no built-in copy button under Claude's responses in the VS Code extension / IDE integration. Selecting and copying text manually works, but a dedicated "copy" icon (similar to what Zed's assistant panel offers) would make it faster to grab a full response without manual selection.

### Proposed Solution

Feature request: add a copy-to-clipboard button under each assistant message.

### Alternative Solutions

_No response_

### Priority

Critical - Blocking my work

### Feature Category

CLI commands and flags

### Use Case Example

_No response_

### Additional Context

_No response_

Start by locating the VS Code extension or IDE integration that renders assistant responses and inspect how message actions are currently handled. Done means each assistant message has a copy-to-clipboard control that copies the full response; verify this across multiple responses.

Written by the indexing model from the issue text.