Thanks so much for sharing this solution!
Meta’s documentation doesn't make this clear at all, and the error message 133010: The account is not registered is super misleading.
So just to make it crystal clear for anyone else who finds this:
For future readers:
Having your number “verified” in Business Manager does NOT mean it’s registered in the API.
You must call:
POST https://graph.facebook.com/v18.0/\<PHONE_NUMBER_ID>/register
with a 6-digit PIN and your access_token.
If you don’t do this, you’ll keep getting the dreaded 133010 error forever.
Thanks again — you saved my sanity (and possibly what's left of my weekend 😅).