[BUG] Supabase HTTP MCP connected in CLI but tools not available in VSCode extension

Preflight Checklist

• [x] I have searched existing issues and this hasn't been reported yet
• [x] This is a single bug report (please file separate reports for different bugs)
• [x] I am using the latest version of Claude Code

What's Wrong?

Supabase MCP shows as ✔ connected when running /mcp in the CLI, but its tools are not exposed in the VSCode extension conversation context.

What Should Happen?

Supabase MCP tools should be available in VSCode extension conversations, same as in the CLI.

Error Messages/Logs

Steps to Reproduce

1. Add an HTTP-type MCP to a project in ~/.claude.json:

"mcpServers": {
  "supabase": {
    "type": "http",
    "url": "https://mcp.supabase.com/mcp"
  }
}

2.Open the project in VSCode with the Claude Code extension
3.Start a new conversation
4.Run /mcp in the CLI — Supabase shows ✔ connected
5.Supabase tools are absent from the conversation; only claude.ai MCPs are available

### Claude Model

Sonnet (default)

### Is this a regression?

I don't know

### Last Working Version

_No response_

### Claude Code Version

2.1.74

### Platform

Anthropic API

### Operating System

Windows

### Terminal/Shell

VS Code integrated terminal

### Additional Information

Note: this is different from #[24770] where /mcp didn't display servers but tools still worked. In my case, the tools are genuinely absent — they do not appear in the available tools list at all.

Environment

OS: Windows 11 Home
Claude Code version: 2.1.74
VSCode extension: official marketplace install
MCP type: http