[MCP]firebase login --reauth when claude call MCP tools

## Summary

├── firebase-tools@15.18.0
├── firebase@12.13.0

Available Project Aliases (format: '[alias]: [projectId]'):

No firebase.json file was found.

If this project does not use Firebase services that require a firebase.json file, no action is necessary.

If this project uses Firebase services that require a firebase.json file, the user will most likely want to:

a) Change the project directory using the 'firebase_update_environment' tool to select a directory with a 'firebase.json' file in it, or
b) Initialize a new Firebase project directory using the 'firebase_init' tool.

Confirm with the user before taking action.

## Bug Info

- **Affected Tool(s):**
- MCP all tools
- **MCP Client:**
- Claude CLI
- **Operating System:**
- MacOS 26.5

### Steps to Reproduce

Start by reproducing the `firebase login --reauth` behavior when Claude CLI invokes Firebase MCP tools, using the reported firebase-tools@15.18.0 and firebase@12.13.0 versions on macOS. Trace the login and MCP tool entry points involved; done means MCP tool calls no longer require an unexpected reauthentication or otherwise fail during authentication.

Written by the indexing model from the issue text.