Model switcher dropdown in chat UI (like Continue.dev)

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

Switching models in Claude Code requires manually editing ANTHROPIC_MODEL in
~/.claude/settings.json and restarting the session. There is no way to switch
models mid-conversation from the UI. This breaks flow, especially when using a
proxy (e.g. LiteLLM) with multiple model aliases configured.

Proposed Solution

Add an inline model picker dropdown in the chat input bar — similar to Continue.dev's
model switcher. The dropdown should list available models (from settings or the proxy)
and allow switching mid-session without restarting. The currently active model should
be clearly displayed (as it already is in the status bar).

Alternative Solutions

Currently editing ~/.claude/settings.json manually and restarting.
The /config command partially helps but still requires a restart.

Priority

Low - Nice to have

Feature Category

Interactive mode (TUI)

Use Case Example

1. Start a session with Claude Opus 4.7 for a complex refactoring task
2. Switch to a faster/cheaper model (e.g. Haiku) for simple Q&A mid-session
3. Switch back to Opus for the next implementation step

— All without leaving the chat or editing files

Additional Context

_No response_