[FEATURE] One-click copy/export conversation history

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

In the VSCode extension, there's no way to quickly copy or export
the full conversation history. The only option is manually selecting
text in the chat panel, which is tedious for long sessions.

I frequently need to extract conversation content to create handoff
documents and share context across sessions.

Proposed Solution

A button or keyboard shortcut to copy/export the full conversation
to clipboard or file (e.g. Markdown format).

Alternative Solutions

Currently I manually select and copy text from the chat panel,
which is impractical for long sessions (50+ messages).

Priority

Low - Nice to have

Feature Category

Interactive mode (TUI)

Use Case Example

1. I use Claude Code in VSCode for multi-step tasks (code review, refactoring, debugging)
2. After a productive session, I want to save the conversation as a handoff document for the next session
3. Currently I have to manually select all text in the chat panel — often 100+ messages
4. A "Copy conversation" button or Ctrl+Shift+C shortcut would let me paste it into a .md file in seconds

Additional Context

Both CLI and VSCode extension would benefit from this.
Markdown output format would be ideal.