I just found out the issue.
Business-initiated calling is not available for U.S. phone numbers in the Meta (WhatsApp Business Cloud API).

That’s why the API returns:

(#138013) Business-initiated calling is not available

So if you're testing with a U.S. WhatsApp number (starting with +1), you won’t be able to send call permission requests or start business-initiated calls.

Switching to a phone number from a supported country resolves the issue.