[FEATURE] Add "Copy Response" button to VS Code Extension

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

The Claude web/desktop app has a convenient "Copy" button for each response that preserves markdown formatting. This feature is missing in the VS Code Claude Code extension.

• No copy button available for assistant responses in the VS Code extension
• Manual text selection and copying often loses or corrupts formatting
• Code blocks have individual copy buttons, but there's no way to copy an entire response with formatting intact

Proposed Solution

• Add a "Copy" button (similar to the Claude web app) that appears on hover or is always visible for each assistant response
• Clicking the button should copy the entire response with markdown formatting preserved
• This would allow users to easily paste responses into documentation, notes, or other markdown-compatible editors

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Developer tools/SDK

Use Case Example

When Claude provides formatted responses with:

• Code blocks
• Bullet points
• Headers
• Links
• Mixed content

Users often need to copy these responses for documentation, sharing, or reference. Currently, this requires manual selection which is error-prone and loses formatting.

Additional Context

The Claude web app already has this feature, so the functionality exists - it just needs to be added to the VS Code extension interface.