Allow disabling claude.ai MCP connector sync to Claude Code

Problem

MCP servers configured on claude.ai automatically sync to Claude Code on every session start. There is no way to disable this behavior, and disabling individual synced servers does not persist across sessions.

Use case

Users who have locally configured MCP servers (e.g., Exa via ~/.claude.json) with custom skills and workflows built around those tool names. When the claude.ai version of the same MCP syncs down, it creates conflicts and clutter because it appears under a different name (e.g., "claude.ai Exa" vs the local "exa"). This means duplicate tools with different names, which breaks skill references and adds noise to the tool list.

Current behavior

• claude.ai MCP connectors sync to Claude Code automatically on every session start.
• Disabling these servers via the /mcp menu works for the current session only. They reappear on every restart.
• The feature is controlled by the tengu_claudeai_mcp_connectors statsig gate with no user-facing toggle to opt out.

Requested solution

A user-facing setting to disable the sync entirely. For example:

// ~/.claude/settings.json
{
  "syncCloudMcpServers": false
}

At minimum, persist the disabled state set via /mcp across sessions so users do not have to re-disable on every restart.

Context

Users need claude.ai MCPs for web chat but do NOT want them in their local Claude Code environment. These are different contexts with different needs — web chat may benefit from cloud-hosted MCP connectors, while a local dev environment has its own carefully configured MCP servers with tool names that skills and workflows depend on. Forcing the cloud versions into the local environment creates unnecessary conflicts.