Gmail MCP: Gmail MCP API confirmed enabled, still "caller does not have permission" (follow-up to #77210)

Follow-up to #77210 (closed/locked before I could reply with the requested checks).

@localden asked us to check, on the GCP project that owns the OAuth client for our custom-added Gmail MCP server:
1. Whether the **Gmail MCP API** service (separate from the standard Gmail API) is enabled.
2. Whether the project needs to be **allowlisted**, since that endpoint is in developer preview.

Results:
- **Gmail MCP API service:** confirmed enabled. In the GCP project (`anchorhr`), APIs & Services → Enabled APIs shows "Gmail MCP API" alongside the standard Gmail-adjacent services (23 enabled APIs total). So it's not a missing-service-enablement issue.
- **Standard Gmail API:** also enabled, as noted in the original issue.

That leaves the allowlisting question, which isn't visible or checkable from the console on our end. **Could you confirm whether project `anchorhr` is allowlisted for the Gmail MCP developer-preview endpoint (`gmailmcp.googleapis.com`)?**

For reference, the failure is unchanged from the original report: OAuth completes, a token is issued, `tools/list` succeeds and returns the full tool set, but the first actual tool call (e.g. `list_labels`) returns "The caller does not have permission." Happy to provide the raw request/response for a specific call if that helps narrow down whether it's a scope-dropping issue on our end vs. an allowlisting gap — let me know what you'd like to see (redacting the token itself, obviously).

Full original context, environment details, and timeline: #77210

No repository file, test, or entry point is identified. Start with issue #77210 and the reported OAuth, tools/list, and list_labels flow; use the requested raw request/response details to distinguish scope handling from project allowlisting. Done means confirming whether project anchorhr is allowlisted for gmailmcp.googleapis.com or identifying the required next diagnostic.

Written by the indexing model from the issue text.