MCP servers added via CLI don't sync to VS Code terminal UI

Bug Description

MCP servers added via claude mcp add in the CLI are not visible in the VS Code extension's "Manage MCP Servers" UI. The two environments maintain completely separate MCP server lists with no synchronization.

Steps to Reproduce

1. Open Claude Code in VS Code terminal
2. Run: claude mcp add --transport http stripe https://mcp.stripe.com --header "Authorization:Bearer sk_xxx"
3. Verify in CLI: claude mcp list → shows server as connected
4. Open VS Code extension's "Manage MCP Servers" dialog → server is NOT listed

Expected Behavior

MCP servers added via CLI should appear in the VS Code extension's MCP management UI (and vice versa), since they share the same .claude.json config file.

Actual Behavior

• CLI (claude mcp list): Shows figma, pencil, stripe — all 3 connected
• VS Code UI ("Manage MCP Servers"): Shows a completely different list (claude.ai Google Calendar, Gmail, Indeed, Figma, Asana, supabase, pencil) — stripe is missing

The CLI reads from C:\Users\<user>\.claude.json (local config scoped to project), while the VS Code extension appears to manage its own separate server list.

Environment

• OS: Windows 11 Home (10.0.26200)
• Claude Code version: Latest (installed via npm)
• VS Code: Latest
• Shell: bash (Git Bash)

Config Locations

The CLI stores MCP config in:

• User config: C:\Users\<user>\.claude.json
• Project config: <project>\.mcp.json
• Local config: C:\Users\<user>\.claude.json [project: <project-path>]

The VS Code extension does not appear to read the local project-scoped config from the CLI.

Impact

Users who configure MCP servers via CLI expect them to work in the VS Code terminal UI and vice versa. The current behavior forces users to configure servers twice — once in CLI and once in VS Code — which is confusing and error-prone.