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

Preflight Checklist

• [x] I have searched existing requests 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_